1. The Long Rambling Philosophical Introduction (*but please
read it anyway)
2. How to Use Advanced Algebra II
3. Functions
1. Introduction
. The Function Game
. "The Real World"
. Algebraic Generalizations
. Graphing
. Permutations
. lest Preparation
. Lines
. Composite Functions
10. Inverse Functions
4. Inequalities and Absolute Values
1. Introduction
2. Inequalities
3. Inequality Word Problems
4. Absolute Value Equations
5. Absolute Value Inequalities
6. Graphing Inequalities and Absolute Values
5. Simultaneous Equations
1. Introduction to Simultaneous Equations
2. Simultaneous Equations
3. The "Generic" Simultaneous Equation
6. Quadratics
1. Introduction
. Multiplying Binomials
. Factoring
. Introduction to Quadratic Equations
. Completing the Square
WON AU BW WN
wi BW N
. The "Generic" Quadratic Equation
. Graphing Quadratic Functions
. Graphing Quadratic Functions II
. Solving Problems by Graphing Quadratic Functions
10. Quadratic Inequalities
7. Exponents
1. Introduction
2. Rules of Exponents
3. Extending the Idea of Exponents
4. Fractional Exponents
5. “Real Life” Exponential Curves
8. Logarithms
1. Introduction
2. Logarithms -- Properties of Logarithms
3. Logarithms -- Using the Laws of Logarithms
4. So What Are Logarithms Good For, Anyway?
9. Rational Expressions
1. Rational Expressions -- Introduction
2. Rational Expressions -- Rational Expressions
3. Rational Expressions -- Rational Equations
4. Polynomial Division
10. Radicals
1. Introduction
2. Radicals -- A Bunch of Other Stuff About Radicals
3. Radical Equations
11. Imaginary Numbers
1. Introduction
2. Introduction to Imaginary Numbers
3. Complex Numbers
WOeoOnN MD
12. Matrices
1. Matrices -- Introduction
ONAUARWHD
. Matrices -- Identity and Inverse Matrices
. Matrices -- Inverse of the Generic 2x2 Matrix
. Matrices -- Use Matrices for Transformation
. Matrices -- Matrices on the Calculator
. Matrices -- Determinants
9.
Matrices -- Solving Linear Equations
13. Modeling Data with Functions
1.
2.
3.
Introduction
Modeling Data with Functions -- Direct and Inverse
Variation
Modeling Data with Functions -- Calculator Regression
14. Conics
1.
. Distance
. Circles
. Parabolas, Day 1
. Parabolas, Day 2
. Parabolas: From Definition to Equation
. Ellipses
. Conic Sections Guide -- Ellipses: From Definition to
COON MU BW N
a:
Conic Sections Guide -- Introduction
Equation
Hyperbolas
15. Sequences and Series
1.
Om
Sequences and Series Guide -- Prerequisites
Sequences and Series Guide -- Arithmetic and Geometric
Sequences
. Sequences and Series Guide -- Series and Series Notation
. Sequences and Series Guide -- Arithmetic and Geometric
Series
. Sequences and Series Guide -- Proof by Induction
6. Sequences and Series Guide -- Extra Credit
16. Probability
1. Probability -- Tree Diagrams
2. Probability -- Introduction to Probability
3. Trickier Probability Problems
4. Permutations
5. Probability -- Combinations
The Long Rambling Philosophical Introduction (*but please read it
anyway)
What you’re holding in your hand is much closer to a set of detailed lesson
plans than to a traditional textbook. As you read through it, your first
reaction may be “Who does he think he is, telling me exactly what to say
and when to say it?”
Please don’t take it that way. Take it this way instead.
Over a period of time, I have developed a set of in-class assignments,
homeworks, and lesson plans, that work for me and for other people who
have tried them. If I give you the in-class assignments and the homeworks,
but not the lesson plans, you only have * of the story; and it may not make
sense without the other third. So instead, I am giving you everything: the in-
class assignments and the homeworks (gathered together in the student
book), the detailed explanations of all the concepts (the other student book),
and the lesson plans (this document). Once you read them over, you will
know exactly what I have done.
What do you do then? You may choose to follow my plan exactly, for a
number of reasons—because it worked for me, or because it looks like a
good plan to you, or just because you have enough other things to do
without planning a lesson that I’ve already planned. On the other hand, you
may choose to do something quite different, that incorporates my ideas in
some form that I never imagined. This book is not a proscription, in other
words, but a resource.
OK, with that out of the way...suppose you decide that you do want to
follow my plan, exactly or pretty closely. Here’s what you do.
e Right now, you read this whole introduction—despite the title, it really
does contain useful information about these materials.
¢ Before beginning each new unit, you read my “conceptual
explanation” of that unit, so you know what I’m trying to achieve.
e Each day before class, you carefully read over my lesson plan (in this
document), and the in-class assignment and homework (in the student
book), so you know what I’m doing and why I’m doing it.
A Typical Day in Mr. Felder’s Class (...and why you care)
At the risk of repeating myself, let me emphasize—I’m not trying to insult
you by suggesting that my way is the only right way to run a class. But it
will help you understand these materials if you understand how I use them.
I begin each day by taking questions on last night’s homework. I answer
any and all questions. This may take five minutes, or it may take the entire
class period: I don’t stop until everyone is perfectly comfortable with last
night’s homework.
Why is that so important? Because, very often, the homework introduces
new concepts that the students have never seen in class before. For
instance, very early in the first unit, I introduce the idea of “permuting”
graphs: for instance, if you add 3 to any function, the graph moves up by
three units. This concept never comes up in class, in any form—it is
developed entirely on a homework. So it’s vitally important to debrief them
the next day and make sure that they got, not only the right answers, but the
point.
After the homework is covered, I begin a new topic. This is almost
(almost!) never done in a long lecture. Sometimes it happens in a class
discussion; sometimes it happens in a TAPPS exercise (more on that when
we do our first one); most often, it happens in an in-class assignment.
These assignments should almost always be done in pairs or groups of
three, very rarely individually. They generally require pretty high-level
thinking. On a good day I can hear three or four heated arguments going on
in different groups. Most of my class time is spent moving between
different groups and helping them when they are stuck. In general, there is
some particular point I want them to get from the exercise, and they will
need that point to do the homework—so a lot of my job in class is to make
sure that, before they leave, they got the point.
Timing
If you read through this entire document (which I do not recommend at one
sitting), you get the illusion that I have everything planned down to the day.
If I say “do this assignment in class, then do this homework,” they had
better get that done in one day, or they will fall irretrievably behind.
Well, suppose you add it all up that way. Every “1-day assignment” (with
homework) counts as one day, and what the heck, let’s allocate two days for
every test (one day for preparation, using the “Sample Test”—and one day
for the actual test). If you add it up that way, you will get a total of 91 days,
or thereabouts. There are 180 days in the school year.
So what does that mean? Does it mean you will be done in one semester?
No, of course not. It means, take your time and do it right.
For one thing, I believe in building in a lot of time for review. Ideally, two
weeks before mid-terms and another two weeks before finals. (What I do
during this time is cover one topic a day, with the students teaching each
class.)
But even leaving that aside, one apparent day’s worth of material will
sometimes take you two days to get through. You spend the whole day
reviewing last night’s homework and you don’t even get to the new
assignment. Or, you get to the end of the class and you realize that most of
the groups are only half-way through the in-class assignment. Don’t rush it!
It’s much more important to get today’s concept, and really make sure
everyone has it, then to rush on to tomorrow. The way I see it, you have
three reasonable choices.
1. If most of the class is mostly finished with the in-class assignment, it
may make sense to say “Finish the in-class assignment tonight, and
also do the homework.”
2. If most of the class is only half-way done, it may make sense to say
“Finish the in-class assignment tonight, and we will do the homework
in class tomorrow.” This puts you a half-day “behind” which is fine.
However, some in-class assignments really cannot be done at
home...they require too much group work or help from you. So...
3. Sometimes you just say “We’ll finish the in-class assignment
tomorrow.” This puts you one day “behind” which is also fine.
Of course you need to pace yourself. But do it by tests, not by days. There
are sixteen tests. If you are going at a clip that will get you through more or
less that many tests by the end of the year, you’re doing fine. And even that
isn’t exact—of course, some units will take longer than others. Personally, I
would much rather skip the unit on Conics (the last unit) entirely, than lose
the entire class by trying to rush through Exponents. (However, in real life,
I do make it through the entire syllabus.)
Tests
At the end of every unit I have a “Sample Test.” This is for the students’
benefit as much as for yours: it makes a great study guide and/or
homework. If you say “The homework tonight is the sample test. Tomorrow
we will go over any questions you have on the sample test, and on the topic
in general—that will be your last chance to ask me questions! The next day
will be our actual test,” then you are giving the students a great chance to
bone up before the test. Doing this has dramatically improved my classes.
So what about your actual test? Of course, you may (or may not) want to
base your test on mine. In that case, however, be careful about timing—
some of my “Sample Tests” are actually too long to be a real test. But they
are made up of actual questions that I have used on actual tests in the past—
and in any case, calling them “Sample Tests” gets students’ attention better
than calling them “Review Questions.”
By the way, although I do not generally recommend using exactly my
questions—you want to change the numbers at least—it is sometimes OK to
use exactly my extra credit. Even if they just did it, it often has enough real
learning in it that it is worth giving them a few points if they took the time
to look it over and/or ask about it.
How to Use Advanced Algebra II
This module contains a table of every module within the three books of
Kenny Felder's course on "Algebra II", with links to the modules.
Over a period of time, I have developed a set of in-class assignments,
homeworks, and lesson plans, that work for me and for other people who
have tried them. The complete set comprises three separate books that work
together:
e The Homework and Activities Book contains in-class and homework
assignments that are given to the students day-by-day.
e The Concepts Book provides conceptual explanations, and is intended
as a reference or review guide for students; it is not used when
teaching the class.
e The Teacher's Guide provides lesson plans; it is your guide to how I
envisioned these materials being used when I created them (and how I
use them myself).
Instructors should note that this book probably contains more information
than you will be able to cover in a single school year. I myself do not teach
from every chapter in my own classes, but have chosen to include these
additional materials to assist you in meeting your own needs. As you will
likely need to cut some sections from the book, I strongly recommend that
you spend time early on to determine which modules are most important for
your state requirements and personal teaching style.
One more warning is important: these materials were designed for an
Advanced Algebra II course. For such a course, I hope this will provide
you with ready-to-use textbook and lesson plans. If you are teaching a
Standard or Remedial-level course, these materials will still be useful, but
you will probably have to cut or reduce some of the most conceptual
material, and supplement it with more drill-and-practice than I provide.
The following table of contents provides a list of topics covered in this
course with links to each module. You can use these links to move between
the books or to jump ahead to any topic.
Activities
Conceptual &
Explanations Homework
[col10624] [col10686]
Functions
Introduction
Function [m18192]
Concepts —— =
What is a
Variable? Wes
What is a
Function? fob
The Rule of [m19190]
Consistency —
Four Ways to
Represent a [m18195]
Function
Domain and
Range [m18191]
The Function [m19121]
Game
The Function
Game: [m19125]
Introduction
Teacher's
Guide
[col10687]
[mi19325)
[m19342]
The Function
Game: Leader's
Sheet
The Function
Game: Answer
Sheet
Functions in the
Real World
Homework:
Functions in the
Real World
Function
Notation
Algebraic
Generalizations
Homework:
Algebraic
Generalizations
Graphing
Homework:
Graphing
Conceptual
Explanations
[col10624]
[m18193]
[m18188]
[m18186]
[m18196]
Activities
&
Homework
[col10686]
[m19126]
[m19124]
[m19115]
[m19114]
[m19108]
[m19116]
Teacher's
Guide
[col10687]
[m19331]
[m19332]
[m19334]
Horizontal and
Vertical
Permutations
Homework:
Horizontal and
Vertical
Permutations I
Homework:
Horizontal and
Vertical
Permutations II
Sample Test:
Functions I
Lines
Homework:
Graphing Lines
Composite
Functions
Homework:
Composite
Functions
Inverse
Functions
Conceptual
Explanations
[col10624]
[m18197]
[m18187]
[m18198]
Activities
&
Homework
[col10686]
[m19110]
[m19119]
[3852 |
[m19122]
[m19113]
[m19118]
[m19109]
[m19107]
[m19112]
Teacher's
Guide
[col10687]
[m19339]
[m19340]
[m19337]
[m19333]
[m19336]
Conceptual
Explanations
[col10624]
Homework:
Inverse
Functions
TAPPS Exercise:
How do I Solve
That For y?
Sample Test:
Functions II
Inequalities and
Absolute Values
Introduction
Inequalities [m18205]
Homework:
Inequalities
Inequality Word
Problems
Absolute Value [m18201]
Equations
Homework:
Absolute Value
Equations
Activities
&
Homework
[col10686]
[m19120]
[m19123]
[m19117]
[m19158]
[m19154]
[m19163]
[m19148]
[m19151]
Teacher's
Guide
[col10687]
[m19432]
[m19430]
[m19428]
[m19426]
Absolute Value
Inequalities
Homework:
Absolute Value
Inequalities
Graphing
Absolute Values
Graphing
Inequalities
Graphing
Inequalities and
Absolute Values
"Piecewise
Functions" and
Absolute Value
Homework:
Graphing
Inequalities and
Absolute Values
Sample Test:
Inequalities and
Absolute Values
Activities
Conceptual &
Explanations Homework
[col10624] [col10686]
[m18207] [m19151]
[m19155]
[m18199]
[m18208]
[m19150]
[m18200]
[m19153]
[m19166]
Teacher's
Guide
[col10687]
[m19431]
[m19433]
Simultaneous
Equations
Introduction to
Simultaneous
Equations
Distance, Rate,
and Time
Simultaneous
Equations by
Graphing
Homework:
Simultaneous
Equations by
Graphing
Substitution
Elimination
Special Cases
Word Problems
Using Letters as
Numbers
Conceptual
Explanations
[col10624]
[m18211]
[m18209]
[m18211]
[m18215]
[m18213]
[m18210]
[m18214]
Activities
&
Homework
[col10686]
[m19288]
[m19291]
[m19291]
Teacher's
Guide
[col10687]
[m19497]
Simultaneous
Equations
Homework:
Simultaneous
Equations
The "Generic"
Simultaneous
Equation
Sample Test: 2
Equations and 2
Unknowns
Quadratics
Introduction
Multiplying
Binomials
Homework:
Multiplying
Binomials
Factoring
Conceptual
Explanations
[col10624]
[m18224]
[m18227]
Activities
&
Homework
[col10686]
[m19293]
[m19289]
[m19294]
[m19292]
[m19247]
[m19253]
[m19243]
Teacher's
Guide
[col10687]
[m19498]
[m19499]
[m19469]
[m19472]
[m19466]
Solving
Quadratic
Equations by
Factoring
Homework:
Factoring
Expressions
Introduction to
Quadratic
Equations
Homework:
Introduction to
Quadratic
Equations
Solving
Quadratic
Equations by
Completing the
Square
Completing the
Square
Homework:
Completing the
Square
Activities
Conceptual &
Explanations Homework
[col10624] [col10686]
[m18222]
[m19248]
[m19246]
[m19251]
[m18217]
[m19242]
[m19249]|
Teacher's
Guide
[col10687]
[m19470]
[m19465]
The Quadratic
Formula
The "Generic"
Quadratic
Equation
Homework:
Solving
Quadratic
Equations
Sample Test:
Quadratic
Equations I
Different Types
of Solutions to
Quadratic
Equations
Graphing
Quadratic
Functions
Graphing
Quadratic
Functions I]
Conceptual
Explanations
[col10624]
[m18231]
[m18216]
[m18228]
Activities
&
Homework
[col10686]
[m19262]
[m19256]
[m19259]
[m19245]
[m19244]
Teacher's
Guide
[col10687]
[m19480]
[m19468]
[m19467]
Homework:
Graphing
Quadratic
Functions I]
Solving
Problems by
Graphing
Quadratic
Equations
Homework:
Solving
Problems by
Graphing
Quadratic
Equations
Quadratic
Inequalities
Homework:
Quadratic
Inequalities
Sample Test:
Quadratics II
Exponents
Introduction
Conceptual
Explanations
[col10624]
[m18220]
[m18230]
Activities
&
Homework
[col10686]
[m19250]
[m19260]
[m19255]
[m19257]
[m19254]
[m19258]
Teacher's
Guide
[col10687]
[m19479]
[m19473]
[m19325]
Exponent
Concepts
Laws of
Exponents
Zero, Negative
Numbers, and
Fractions as
Exponents
Exponential
Curves
Rules of
Exponents
Homework:
Rules of
Exponents
Extending the
Idea of
Exponents
Homework:
Extending the
Idea of
Exponents
Conceptual
Explanations
[col10624]
[m18232]
[m18235]
[m18234]
[m18233]
Activities
&
Homework
[col10686]
[m19104]
[m19101]
[m19096]
[m19098]
Teacher's
Guide
[col10687]
[m19327]
[m19328]
Fractional
Exponents
Homework:
Fractional
Exponents
"Real Life"
Exponential
Curves
Homework:
"Real Life"
Exponential
Curves
Sample Test:
Exponents
Logarithms
Logarithm
Concepts
Logarithms
Explained by
Analogy to
Roots
Conceptual
Explanations
[col10624]
[m18242]
[m18236]
Activities
&
Homework
[col10686]
[m19097]
[m19100]
[m19103]
[m19102]
[m19105]
Teacher's
Guide
[col10687]
[m19322]
[m19329]
Conceptual
Explanations
[col10624]
Rewriting
Logarithm
Equations as [m18241]
Exponent
Equations
The Logarithm
Defined as an [m18240]
Inverse Function
Introduction
Homework:
Logs
Properties of
Logarithms [m18239]
Homework:
Properties of
Logarithms
Using the Laws
of Logarithms
Common
Logarithms bm 8237)
Graphing
Logarithmic [m18238]
Functions
Activities
&
Homework
[col10686]
[m19175]
[m19176]
[m19269]
[m19177]
[m19184]
Teacher's
Guide
[col10687]
[m19436]
[m19438]
[m19440]
So What Are
Logarithms
Good For,
Anyway?
Homework: So
What Are
Logarithms
Good For,
Anyway?
Sample Test
Rational
Expressions
Introduction
Rational
Expressions
Concepts
Simplifying
Rational
Expressions
Multiplying
Rational
Expressions
Conceptual
Explanations
[col10624]
[m18304]
[m18296]
[m18301]
Activities
&
Homework
[col10686]
[m19181]
[m19268 |
[m19180]
Teacher's
Guide
[col10687]
[m19439]
[m19486]
Adding and
Subtracting
Rational
Expressions
Rational
Expressions
Homework:
Rational
Expressions
Rational
Equations
Homework:
Rational
Expressions and
Equations
Dividing
Polynomials
Sample Test:
Rational
Expressions
Radicals
Radical
Concepts
Conceptual
Explanations
[col10624]
[m18303]
[m18302]
[m18299]
[m18244]
Activities
&
Homework
[col10686]
[m19278]
[m19275]
[m19279]
[m19277]
[m19276]
[m19274]
Teacher's
Guide
[col10687]
[m19488]
[m19489]
[m19487]
Conceptual
Explanations
[col10624]
Radicals (*aka
Roots)
Properties of
71
Radicals ee
Radicals and
Exponents
Some Very
Important
Generalizations
Simplifying
Radicals eat
Introduction
Homework:
Radicals
A Bunch of
Other Stuff
About Radicals
Homework: A
Bunch of Other
Stuff About
Radicals
Activities
&
Homework
[col10686]
[m19420]
[m19419]
[m19422]
[m19421]
[m19270]
[m19263]
[m19264]
Teacher's
Guide
[col10687]
[m19484]
[m19483]
Conceptual
Explanations
[col10624]
Radical [m18273]
Equations ——
Homework:
Radical
Equations
Sample Test:
Radicals
Imaginary
Numbers
Introduction
Imaginary
Numbers
Concepts
[m18285]
Playing with i [m18286]
Introduction to
Imaginary
Numbers
Imaginary
Numbers
Homework:
Imaginary
Numbers
Activities
&
Homework
[col10686]
[m19272]
[m19271]
[m19273]
[m19129]
[m19130]
Teacher's
Guide
[col10687]
[m19485]
[m19424]
[m21990]
Conceptual
Explanations
[col10624]
Complex
Numbers LmLB282]
Equality and
Inequality in
Comples [m18283]
Numbers
Homework:
Complex
Numbers
Quadratic
Equations and
Complex [m18288]
Numbers
Me, Myself, and
the Square Root
of i
The Many Merry
Cube Roots of -1
Homework:
Quadratic
Equations and
Complex
Numbers
Activities
&
Homework
[col10686]
[m19128]
[m19132]
[m19134]
[mi973 1]
[m19127]
Teacher's
Guide
[col10687]
[m19423]
[m19425]
A Few "Extras
For Experts"
Thoughts on
Imaginary
Numbers
Sample Test:
Complex
Numbers
Matrices
Matrices
Introduction to
Matrices
Homework:
Introduction to
Matrices
Multiplying
Matrices
Multiplying
Matrices I
Homework:
Multiplying
Matrices I
Conceptual
Explanations
[col10624]
[m18284]
[m18311]
[m18291]
Activities
&
Homework
[col10686]
[m19133]
[m19206]
[m19205]
[m19207]
[m19196]
Teacher's
Guide
[col10687]
[m19445]
[m19448]
Multiplying
Matrices II
Homework:
Multiplying
Matrices II
The Identity
Matrix
The Inverse
Matrix
The Identity and
Inverse Matrices
Homework: The
Identity and
Inverse Matrices
The Inverse of
the Generic 2x2
Matrix
Using Matrices
for
Transformations
Conceptual
Explanations
[col10624]
[m18293]
[m18294]
Activities
&
Homework
[col10686]
[m19208]
[m19201]
[m19213]
[m19194]
[m19214]
[m19221]
Teacher's
Guide
[col10687]
[m19449]
[m19443]
[m19446]
[m19451]
Homework:
Using Matrices
for
Transformations
Sample Test:
Matrices I
Matrices ona
TI-83 or TI-84
Calculator
Matrices on the
Calculator
Homework:
Calculators
Determinants
Homework:
Determiners
Solving Linear
Equations
Homework:
Solving Linear
Equations
Conceptual
Explanations
[col10624]
[m18290]
[m18289]
[m18292]
Activities
&
Homework
[col10686]
[m19190]
[m19210]
[m19188]
[m19193]
[m19212]
[m19204]
Teacher's
Guide
[col10687]
[m19447]
[m19442]
[m19450]
Sample Test:
Matrices II
Modeling Data
with Functions
Introduction
Data Modeling
Concepts
Direct and
Inverse Variation
Direct Variation
Homework:
Inverse Variation
Homework:
Direct and
Inverse Variation
Finding a Linear
Function For
Any Two Points
Activities
Conceptual &
Explanations Homework
[col10624] [col10686]
[m19209]|
[m18277]
[m18281]
[m19228]
[m19227]
[m19225]
[m18278]
Teacher's
Guide
[col10687]
[m19454]
[m19452]
Finding a
Parabolic
Function For
Any Three
Points
From Data
Points to
Functions
Homework:
From Data
Points to
Functions
Calculator
Regression
Homework:
Calculator
Regression
Sample Test:
Modeling Data
With Functions
Conics
Introduction
Conic Concepts
Conceptual
Explanations
[col10624]
[m18279]
[m18280]
[m18265]
Activities
&
Homework
[col10686]
[m19224]
[m19232]
[m19231]
[m19222]
Teacher's
Guide
[col10687]
[m19453]
[m19307]
A Mathematical
Look at Distance
Distance
Homework:
Distance
Circles
All the Points
Equidistant from
a Given Point
Homework:
Circles
Parabolas
All the Points
Equidistant from
a Point and a
Line
Parabolas: Day 1
Homework:
Vertical and
Horizontal
Parabolas
Conceptual
Explanations
[col10624]
[m18246]
[m18245]
[m18268]
Activities
&
Homework
[col10686]
[m19081]
[m19086]
[m19078]
[m19084]
[m19079]
[m19086]
Teacher's
Guide
[col10687]
[m19299]
[m19298]
[m19315]
Parabolas: Day 2
Parabolas: From
Definition to
Equation
Sample Test:
Distance,
Circles, and
Parabolas
Ellipses
Distance from
this point plus
distance to that
point is Constant
Homework:
Ellipses
Ellipses: From
Definition to
Equation
Hyperbolas
Conceptual
Explanations
[col10624]
[m18247]
[m18249]
Activities
&
Homework
[col10686]
[m19092 |
[m19094]
[m19083]
[m19088]
[m19095]
Teacher's
Guide
[col10687]
[m19313]
[m19311]
[m19303]
[m19305]
[m19306]
Distance from
this point minus
distance from
that point is
constant
Homework:
Hyperbolas
A Brief Recap:
How Do You
Tell What Shape
It Is?
Sample Test:
Conics 2
(Ellipses and
Hyperbolas)
Sequences and
Series
Prerequisites
Sequences
Arithmetic and
Geometric
Sequences
Activities
Conceptual &
Explanations Homework
[col10624] [col10686]
[m19082]
[m19089]
[m18270]
[m19093]
[m19076]
[m19285]
Teacher's
Guide
[col10687]
[m19495]
[m19490]
Homework:
Arithmetic and
Geometric
Sequences
Series
Series and Series
Notation
Homework:
Series and Series
Notation
Arithmetic and
Geometric
Series
Homework:
Arithmetic and
Geometric
Series
Proof by
Induction
Homework:
Proof by
Induction
Extra Credit
Activities
Conceptual &
Explanations Homework
[col10624] [col10686]
[m19284]
[m19074]
[m19280]
[m19282]|
[m19075]
[m19281]
Teacher's
Guide
[col10687]
[m19491]
[m19494]
[m19492]
[m19493]
Conceptual
Explanations
[col10624]
Sample Test:
Sequences and
Series
Probability
How Many
Groups?
Tree Diagrams
Homework: Tree
Diagrams
Probability [m19073]
Concepts
Introduction to
Probability
Homework: The
Multiplication
Rule
Trickier
Probability
Problems
Activities
&
Homework
[col10686]
[m19283]
[m19236]
[m19234]
[m19237]
[m19233]
Teacher's
Guide
[col10687]
[m19463]
[m19461]
[m19464]
Homework:
Trickier
Probability
Problems
Permutations
Homework:
Permutations
Combinations
Homework:
Permutations
and
Combinations
Sample Test:
Probability
Conceptual
Explanations
[col10624]
[m19072]
[m19071]
Activities
&
Homework
[col10686]
[m19235]
[m19241]
[m19240]
[m19238]
Teacher's
Guide
[col10687]
[m19462]
[m19460]
Introduction
The introduction to the teacher's guide on functions.
This is the most important unit in the year, because it introduces many of
the major “themes” that will run through the entire class. These themes
include:
¢ What is a function?
e How are functions used to model things in the real world?
e What does it mean for two functions to be “equal”?
e How does a functional description f(x) = 32 relate to a graph?
More detailed topics include the vertical line test (the “rule of
consistency”), the “dependent” and “independent” variables, domain and
range, composite functions, and inverse functions.
Also included in this unit is a quick review of graphing lines. It is assumed
that students are mostly familiar with this topic from Algebra I.
The Function Game
An explanation of the function game.
This game is an introduction to the idea of a function.
Begin by breaking the students into groups of three. In each group, one
student is designated as the leader and another as the recorder. (Do this
quickly and arbitrarily: “The shortest person is the leader and the tallest is
the recorder” or some such. Assure them that the roles will rotate.) Go over
the instructions (which are in the student packet): walk through a sample
session, using the function “add five,” to make sure they understand who
does what and what gets written down. In particular, make sure they
understand how “add five” can be represented as “a + 5”: that is, that x is
being used to designate the number that comes in. You also want to mention
domain in particular, and the idea that if you are doing and someone
gives you a O, it is “not in your domain.”
Then they can start. Your job is to circle around, keeping them on task, and
helping students who are stuck by giving hints: “Do you notice anything in
common about all the numbers you’ve gotten back?” or “Why are all
negative numbers outside the domain?” or even “Try a 2 and see what
happens.” Also, at 10 or 15 minute intervals, instruct them to switch roles.
Toward the end of the class period, interrupt briefly to talk about the word
function. What is the leader representing? He is not a number. He is not a
variable. He is a process that turns one number into another. That’s all a
function is—a mechanical process that takes one number in, and spits a
different number back out. Use the analogy of a little machine (you can
draw it) with an input and an output. Functions are going to be the main
focus of the entire year in Algebra II.
If some groups don’t finish all the problems, that’s OK: as long as they did
enough to get the idea. If a group finishes early, tell them to start making up
their own functions—challenge them to stump you!
Homework:
"Homework: The Function Game"
"The Real World"
The teacher's guide to real world problems using functions.
Begin by going over the homework. There are two key points to bring out,
which did not come up in class yesterday.
1. If two functions always give the same answer, we say they are equal.
This just cannot be stressed enough. Put this on the board:
3(a — 5) = 3a — 15
They’ve all seen this. Most of them know it is the “distributive property.”
But do they know what it means? Explain it very carefully. 3(@ — 5) isa
function that says “subtract 5, then multiply by 3.” 3(a@ — 15) is a function
that says “multiply by 3, then subtract 15.” They are very different
processes! But we say they are “equal” because no matter what number you
plug in, they yield the same answer. So in the function game, there is no
possible question you could ask that would tell you if the person is doing
“subtract 5, then multiply by 3” or “multiply by 3, then subtract 5.” Have
the class give you a few numbers, and show how it works—for negative
numbers, fractions, zero, anything.
2. The last problem on the homework brings up what I call the “rule of
consistency.” It is perfectly OK for a function to give the same answer for
different questions. (For instance, x? turns both 3 and -3 into 9.) But it is
not OK to give different answers to the same question. That is, if a function
turns a 3 into a 9 once, then it will always turn a 3 into a 9.
Once those two points are very clearly made, and all questions answered,
you move on to today’s work. Note that there is no “in-class assignment” in
the student book: this is a day for interacting with the students.
Remind them that a function is simply a process—any process—that takes
one number in, and spits a different number out. Then explain that functions
are so important because they model relationships in the real world, where
one number depends on a different number. Give them a few examples like
the following—no math, just a verbal assertion that one number depends on
another:
e The number of toes in class depends on the number of feet in class.
¢ Which, in turn, depends on the number of people in class.
e The number of points you make in basketball depends on how many
baskets you make.
e The amount I pay at the pump depends on the price of gas.
Have them brainstorm in pairs (for 1-2 minutes at most) to come up with as
many other examples as they can. Since this is the first “brainstorming”
exercise in class, you may want to take a moment to explain the concept.
The goal of brainstorming is quantity, not quality. The object is to come
up with as many examples as you can, no matter how silly. But although
they may be silly, they must in this case be valid. “The color of your shirt
depends on your mood” is not a function, because neither one is a number.
“The number of phones in the class depends on the number of computers”
is not valid, because it doesn’t. By the end of a few minutes of
brainstorming and a bit more talk from you, they should be able to see how
easy it is to find numbers that depend on other numbers.
Then—after the intuitive stuff—introduce formal functional notation.
Suppose you get two points per basket. If we let p represent the number of
points, and 6 represent the number of baskets, then p(b) = 2(b). If we say
p(c) = 2c that is not a different function, because they are both ways of
expressing the idea that “p doubles whatever you give it” (relate to the
function game). So if you give it a 6, you get 12 : p(6) = 12. If you give it
a duck, you get two ducks: p(duck) = 2ducks. It doubles whatever you
give it.
Key points to stress:
e This notation, p(b), does not indicate that p is being multiplied by b. It
means that p depends on J, or (to express the same thing a different
way), p is a function of b.
e You can plug any numbers you want into this formula. p(6) = 12
meaning that if you get 6 baskets, you make 12 points. p(2>) = dis
valid mathematically, but in the “real world” you can’t make 25
baskets. Remind them that p is a function—a process—“double
whatever you are given.”
e Also introduce at this point the terminology of the dependent and
independent variables.
e Stress clearly defining variables: not “b is baskets” but “b is the
number of baskets you make.” (“Baskets” is not a number.)
e Finally, talk about how we can use this functional notation to ask
questions. The question “How many points do I get if I make 4
baskets?” is expressed as “What is p(4)?” The question “How many
baskets do I need in order to get 50 points?” is expressed as “
p(b) = 50, solve for 6.”
For the rest of class—whether it is five minutes or twenty—the class should
be making up their own functions. The pattern is this:
1. Think of a situation where one number depends on another. (“Number
of toes depends on number of feet.’’)
2. Clearly label the variables. (t=number of toes, f=number of feet.)
3. Write the function that shows how the dependent variable depends on
the independent variable (t(f) = 5f. )
4. Choose an example number to plug in. (If there are 6 feet,
t(f) = 5(6) = 30. 30 toes.)
Encourage them to think of problems where the relationship is a bit more
complicated than a simple multiplication. (“The area of a circle depends on
its radius, A(r) = tr”),
Homework:
“Homework: Functions in the Real World.”
When going over this homework the next day, one question that is almost
sure to come up is #4g: f(f(a)). Of course, I’m building up to the idea of
composite functions, but there is no need to mention that at this point. Just
remind them that f (anything) = anything” + 2anything + 1. So the
answer to (e) is f (spaghetti) = spaghetti? + 2spaghetti + 1. And the
answer to this one is f(f(x)) = f(x)? + 2f(a) +1. Of course, this can
(and should) be simplified, but the point right now is to stress that idea that
you can plug anything you want in there.
Algebraic Generalizations
A teacher's guide to using algebraic generalizations to arrive at a function.
This is the real fun, for me.
Start by telling the students “Pick a number. Add three. Subtract the number
you started with. You are left with...three!” OK, no great shock and
surprise. But let’s use algebra to express what we have just discovered.
x+3— ax =3. The key is recognizing what that sentence mean. x can be
any number. So when we write z + 3 — x = 3 we are indeed asserting that
if you take any number, add three and then subtract the number, you get
three in the end.
Here’s a harder one. Pick a number, add three, multiply by four, subtract
twelve, divide by the number you started with. Everyone started with
different numbers, but everyone has 4 in the end. Ask the students to find a
generalization to represent that, and see if they can work their way to
4(2+3)—12
x
work with.
= 4. See also if they can guess what number this trick will not
Now, have them work on the in-class assignment “Algebraic
Generalizations,” in groups of three.. Most of the class period should be
spent on this. This is hard!!! After the first couple of problems (which very
directly echo what you already did in class), most groups will need a lot of
help.
Here are some of the answers I’m looking for—lI include this to make sure
that the purpose of the assignment is clear to teachers.
¢ In #3, the object is to get to 27*! = 2 e 2” (or, equivalently,
2 = 2% taal Talk through this very slowly with individual groups.
If you wanted to get from 2!° to 2!! , what would you do? And to get
from 2°° to 2°4 ? And what about 2°° to 21° ? Can you say in English
what we're saying, in general? Now, can you say that in math? etc...
The real goal is to get them to see how, once you have written
27+! — 2 e 2”, you have said in one statement that 2° is twice 2’, and
also that 2° is twice 2%, and also that 2!! is twice 21°, and so on. It is a
“generalization” because it is one statement that represents many
separate facts.
¢ In #4, the object is to get to 22x? = x**?, Again, it will take a lot of
hand-holding. It isn’t important for them to do it entirely on their own.
It is critically important that, by the time they are done, they see how
those numbers lead to that generalization; and how that generalization
leads to those numbers.
e If all that works, they should be able to do #6 and come up with
something like (x 1) (a + 1) = x? — 1 pretty much on their own.
(Or, of course, x? = (a — 1) (a — 1) + 1, which is a bit more
unusual-looking but just as good.)
Homework:
“Homework: Algebraic Generalizations”
Graphing
A teacher's guide to graphing functions.
Make sure, on the homework, that they reached something like
(a = a) (a + a) — x” — a*. Of course, someone may have used
completely different letters; and someone else may have said that
(a = a) (a = a) + a? = x”. Point out that these are just as good—they
look different, but they say the same thing. Finally, show them how they
could have arrived at (x — a) (a +a) = 2 — a? through FOIL.
Once all questions are satisfied, on to graphing. This may actually be two
days worth of material. Don’t rush it! And once again, no in-class
assignment, but a lot of interaction.
Start by putting the following points on the board: (6,5)(3,6)(2,9)(5,7)(1, 10)
(4,8). Ask for a general description—if these represent the number of
pushups you do each day, what’s the trend? Then graph them, and you can
see a definite downward trend with an odd spike in the middle. Moral: we
can see things in shapes that we can’t see in numbers.
Now, let’s jump to the idea of graphing functions. Draw a U-shape, the
equation f (x) = x”, and say “This drawing is the graph of that function—
but what does that mean? What does this drawing actually have to do with
that function?” Get to the point where the following ideas have come out.
Every time you “do” the function, one number goes in and another comes
out. When we graph it, the “in” function is always 2, and the “out” function
is always y—in other words, every time we graph a function, we are
always graphing y = f(x). To put it another way, we are graphing all the
points that have this particular relationship to each other. (Take as much
time as you need on this point.)
Have the students graph |x| (individually at their desks) by plotting points.
This is not intended to teach them about absolute value, but to reinforce the
ideas I just made about what it means to graph a function.
What can we tell by looking at a graph? Draw the graphs of x? and x? on
the board. Talk about the things we can tell about these two functions by
looking at the graphs. They both have one zero; they are similar on the
right, except that x? rises faster; they are completely different on the left;
they both have unlimited domains, but only one has an unlimited range.
(Make sure to connect this back to “domain” and “range” from the function
game!)
Now, draw this on the board.
y = f(x}
5
Time for another...brainstorming exercise! (Remind them that the object is
quantity, not quality!) Each pair of students has to list as many things as it
can tell about this function f(x) by looking at the graph. Key points I want
to bring out are:
e The three zeros. (Talk about the word “zeros.”
e Places where the function is negative and places where it is positive.
e Places where it is increasing and places where it is decreasing. (Talk
about slope!)
e What happens for very low and very high values of x.
e For the experts, odd symmetry: f(—x) = —f(a).
Talk more about domain and range. This is unrestricted in both. We saw
that x” has an unrestricted domain, but a restricted range. Why would any
function have a limited domain? Generally, it is because of the two "thou
shalt not" rules: thou shalt not divide by zero, and thou shalt not take the
square root of a negative number. So, consider the following three
statements.
1. You can't take the square root of a negative number.
2. The domain of the function y=./z is x>0.
3. The graph of y=,/z starts at the y-axis, and goes to the right; it doesn't
go to the left. (Draw it.)
Give them all three of these statements, and see if they can see that they are
all three saying the same thing. Then see if they can generate three
equivalent versions of "You can't divide by zero." ("The domain of the
function y=1/z is all numbers except 0," and "The graph of y=1/a never
touches the y-axis.")
Remind them of the “rule of consistency” that we discussed earlier: a
function can never take one input and generate two different outputs. Ask
them to discuss in pairs, for one minute, how this rule manifests on a graph.
Then have class discussion until you have reached the vertical line test.
Come back to the question we started with: why do we graph things? As we
demonstrated earlier, graphs enable us to see things visually that are very
hard to see in numbers. Draw several different jaggy, shaky graphs, and
suggest that they represent the price of gas—ask for verbal descriptions of
what each one tells us. Talk again about domain, range, positive, negative.
Throw something straight up into the air and catch it. Tell them there is a
function h(t) that represents the height of that object, as a function of time.
(Make sure they get this.) Give them one minute to sketch the graph of that
function. Then show them that it is an upside-down parabola (you don’t
need to use that word). Emphasize that this does not mean the object
traveled in an arc shape: it traveled straight up and down. The horizontal
axis is time and the vertical axis is height.
If you have a bit of extra time, explain how to generate graphs (and set the
window) on the calculator.
Homework:
“Homework: Graphing”
Permutations
A teacher's guide to permutations and functions.
The end of the “Graphing” homework sets this topic up.
Have one person in the class “be” x”. He is allowed to use a calculator; so
you can, for instance, hand him the number 1.7 and he will square it,
producing the point (1.7,2.89).
Another person is 2” + 1. He is not allowed to use a calculator, but he is
allowed to talk to the first person, who is. So if you hand him 1.7, he asks
the first person, who says 2.9, and then he comes back with a 3.9. Make
sure everyone understands what we have just learned: the graph of x? + 1
contains the point (1.7,3.9). Do a few points this way.
Another person is (2 + 1)? with the same rules. So if you give him a 1.7 he
hands a 2.7 to the calculator person. Make sure everyone understands how
this process gives us the point (1.7,7.3).
Talk about the fact that the first graph is a vertical permutation: it messed
with the y-values that came out of the function. It’s easy to understand what
it did. It added 1 to every y-value, so the function went up 1.
The second graph is a horizontal permutation: it messed with the x-values
that went into the function. It’s harder to see what that did: why did
(x + 1)? move to the left? Ask them to explain that.
Now hand them the worksheet “Horizontal and Vertical Permutations I.”
Hand one to each person—they will start in class, but probably finish in the
homework. It’s on the long side.
The next day, talk it all through very carefully. Key points to bring out:
1. What does f(a) + 2 mean? It means first plug a number into f(z),
and then add 2.
2. And what does that do to the graph? It means every y-value is two
higher than it used to be, so the graph moves up by 2.
3. What does f(a + 2) mean? It means first add 2, then plug a number
into f(x).
4. And what does that do to the graph? It means that when x = 3 you
have the same y-value that the old graph had when z = 5. So your
new graph is to the left of the old one.
5. What does all that have to do with our rock? This should be a long-ish
conversation by itself. The vertical and horizontal permutations
represent very different types of changes in the life of our rock.
Suggest a different scenario, such as our old standard, the number of
candy bars in the room as a function of the number of students, c(s).
What scenario would c(s) + 3 represent? How about c(s + 3)?
If you haven’t already done so, introduce graphing on the calculator,
including how to properly set the window. It only takes 5-10 minutes, but is
necessary for the homework.
Now, put the graph of y = x? on the board. We saw what (x + 1)? and
a” + 1 looked like yesterday. What do you think —x? would look like?
How about (x + 2)? — 3?
At some point, during the first or second day, you can come back to the idea
of domain. What is the domain of y = Vx + 3? See if they can see the
answer both numerically (you can plug in 2 = —3 but not = —4) and
graphically (the graph of y = ./z moved three spaces to the left, and its
domain moved too).
Homework:
“Horizontal and Vertical Permutations IT”
Test Preparation
A teacher's guide to preparing a sample test over functions.
Follow up carefully on the homework from the day before. In #8, make sure
they understand that losing money is negative profit. But mostly talk about
#9. Make sure they understand how, and why, each modification of the
original function changed the graph. When we make a statement like
“Adding two to a function moves the graph up by 2” this is not a new rule
to be memorized: it is a common-sense result of the basic idea of graphing a
function, and should be understood as such.
Ask what they think — (x + 1)?~3 would look like, and then show them
how it combines three of the modifications. (The +1 moves it to the left,
the —3 moves it down, and the — in front turns it upside-down. )
Talk about the fact that you can do those generalizations to any function, eg
—|a + 1|-3 (recalling that they graphed absolute value yesterday in class).
Draw some random squiggly f(a) on the board, and have them draw the
graph of f(a) + 2. Then, see if they can do f(x + 2).
Go back to the idea of algebraic generalizations. We talked about what it
means for two functions to be “equal”—what does that look like,
graphically? They should be able to see that it means the two functions have
exactly the same graph. But this is an opportunity for you to come back to
the main themes. If two functions are equal, they turn every x into the same
y. So their graphs are the same because a graph is all the (x,y) pairs a
function can generate!
I also like to mention at this point that we actually use the = sign to mean
some pretty different things. When we say x + 3 = 5 we are asking for
what value of z is this true? Whereas, when we say 2(2 — 7) = 2” — 14
we are asserting that this is true for all values of x. Mention, or ask them
to find, statements of equality that are not true for any value of z (eg
£= 7-4 1),
Now, at this point, you hand out the “sample test.” Tell them this test was
actually used for a past class; and although the test you give will be
different, this test is a good way of reviewing. I have found this technique—
handing out a real test from a previous year, as a review—to be
tremendously powerful. But I have to say a word here about how to use it.
Sometimes I say “Work on it in class if you have time, glance it over
tonight if it helps you study; the test is tomorrow.” And sometimes I say
“This is the homework. Tonight, do the sample test, and also look over all
the materials we have done so far. Tomorrow, we will go over the sample
test and any questions you have on the material, in preparation for the test,
which will be the day after tomorrow.” It all depends on timing, and on how
prepared you think the class is.
Homework:
“Sample Test: Functions I”
I should say a word about #2 here, just to be clear about what I’m looking
for. We have a function c(s). In part (b) I supply s = 20 and ask for c: so
the question in function notation is c(20). (Then you plug a 20 into the
formula and go from there.) In part (c) I supply c = 35 and ask for c(20):
so this question in function notation is c(s) = 35. (Then you set the
formula equal to 35 and solve.) Students have a lot of difficulty with this
asymmetry.
Now give a test of your own on functions I
You may use something very similar to my test, except changing numbers
around. Or you may do something quite different.
I will tell you, as a matter of personal bias, that I feel very strongly about
question #7. Many students will do quite poorly on this question. For
instance, they may give you two variables that do not in fact depend on
each other (number of guitarists and number of drummers). Or they may
give you variables that are in fact constants (let m equal the number of notes
in an octave). The answers don’t have to be complicated, although
sometimes they are—sometimes a very simple answer gets full credit.
(“CDs cost $12 apiece, I spend d dollars on c CDs. d(c) = 12c” is a full
credit answer to parts a-c.) But they have to show that they can clearly
articulate what the variables are, and how they depend on each other. If they
cannot, spend the time to explain why it is wrong, and work them through
correct answers. In my opinion, this skill is the best measure of whether
someone really understands what a function is. And it is a skill like any
other, in the sense that it develops over time and practice.
(Also, I don’t believe in surprises. Tell them in advance that this question
will definitely be on the test—the only thing that will change is the topic.)
Lines
A teacher's guide to the section on lines in preparation for later lectures on
functions.
This is largely review: if there is one thing the students do remember from
Algebra I, it’s that y = ma + 6 and m is the slope and 6 is the y-intercept.
However, we’re going to view this from the viewpoint of linear functions.
Start by giving an example like the following: I have 100 markers in my
desk. Every day, I lose 3 markers. Talk about the fact that you can write a
function m(d) that represents the number of markers I have as a function of
day. It is a linear function because it changes by the same amount every
day. If I lose three markers one day and four the next, there is still a
function m(d), but it is no longer a linear function. (If this is done right, it
sets the stage for exponential functions later: linear functions add the same
amount every day, exponential functions multiply by the same amount
every day. But I wouldn’t mention that yet.)
So, given that it changes by the same amount every day, what do you need
to know? Just two things: how much it changes every day (—3), and where
it started (100). These are the slope and the y-intercept, respectively. So we
can say y = —3a + 100 but I actually prefer to write the 6 first:
y = 100 — 3z. This reads very naturally as “start with 100, and then
subtract 3, x times.”
Hammer this point home: a linear function is one that adds the same amount
every time. Other examples are: I started with $100 and make $5.50 each
hour. (Money as a function of time.) I start on a 40' roof and start piling on
bricks that are - ‘each. (Height as a function of number of bricks.)
Then talk more about slope—that slippery concept that doesn’t tell you how
high the function is at all, but just how fast it’s going up. With a few quick
drawings on the board, show how you can look at a line and guesstimate its
slope: positive if it’s going up, negative if it’s going down, zero for
horizontal. You can’t necessarily tell the difference between a slope of 3
and a slope of 5, but you can immediately see the difference between 3 and
t. Emphasize that when we say “going up” and “going down” we always
mean as you go from left to right: this is a very common source of errors.
Talk about the strict definition of slope. Actually, I always give two
definitions. One is: every time z increases by 1, y increases by the slope.
(Again: if the slope is negative, y decreases.) The other is: for any two
points on the line, the slope is ae “rise over run”). Show that this ratio is
the same whether you choose two points that are close, or two points that
are far apart. Emphasize that this is only true for lines.
Finally, why is it that in y= mz + B, the b is the y-intercept? Because the y
-intercept is, by definition, the value of y when x = 0. If you plug z = 0
into y= mz + b you get y = b.
All this may take all day, or more than one day. Or, it may go very quickly,
since so much of it is review. When you’re done, have them work the in-
class assignment “Lines” in groups.
Homework:
“Homework: Graphing Lines”
Composite Functions
A teacher's guide to composite functions.
OK, it’s getting hard again: this one may take a couple of days. We start
with discussion.
There are a number of ways to look at composite functions. It’s important to
be able to use all of these ways, and to see how they relate.
1. All the way back to the function game. Let one student be the function
Ax + 6, and another student be the function x /2. You give a number to
the first student, who spits out a number at the second student, who
spits out a number back at you. For instance, if you give the first
student a 3, his output is an 18; the second student takes this 18 and
comes out with a 9. Do this for a while until everyone has the hang of
it. See if anyone realizes that what’s going on is, in the end, the
function 2% + 3 is being done to your number.
2. Now, talk about a factory. One box turns garbage into gloop; the next
box turns gloop into shlop; the final box turns shlop into food. Each
box can be represented by a function that says “If this much goes in,
that much goes out.” The entire factory is a gigantic composite
function, where the output of each box is the input of the next, and the
composite function says “If this much garbage goes in, this much food
goes out.” (Draw it!)
3. In general, composite functions come up with this variable depends
on that variable, which in turn depends on the other variable. The
amount of taxes you pay depends on the amount of money you make,
which in turn depends on the number of hours you work. Have them
come up with a few examples. Be very careful to distinguish
composite functions from multivariate functions, e.g. the number of
kids in the class depends on the number of boys, and the number of
girls. That is not a function, because those two variables don’t depend
on each other.
4. Finally, there is the formalism, f(g(x)). Remind them that this is
mechanical. If g(a) = 4% + 6 and f(x) = 2/2, then what is f(g(x))?
Well, f(anything) = anything/2. So f(g(x)) = g(x) /2, which is
(4x + 6)/2 or 2x + 3. Note that this is completely different from
g( f(x))! Take a moment to connect this mechanical process with the
idea of a composite function that you have already discussed.
Now, have them work through the in-class assignment on “Composite
Functions” in groups. Make sure they do all right on #4.
#6 is a build-up to inverse functions, although you don’t need to mention
that. If anyone asks for help, help them see that if h(x) = x — 5, then
h(anything) = anything — 5, so h(i(x)) = i(x) — 5. Soi(x) —5 = 2,
and we can solve this to find i(x) = x + 5.
Homework:
“Homework: Composite Functions”
Inverse Functions
A teacher's guide to inverse functions.
This is definitely two days, possibly three.
Just as with composite functions, it is useful to look at this three different
ways: in terms of the function game, in terms of real world application, and
in terms of the formalism.
e 1Ask a student to triple every number you give him and then add 5. Do
a few numbers. Then ask another student to reverse what the first
student is doing. This is very easy. You give the first student a 2, and
he gives you an 11. Then you give the second student an 11, and he
gives you a 2. Do this a few times until everyone is comfortable with
what is going on. Then ask what function the second student is doing.
With a little time, everyone should be able to figure this out—he is
reversing what the first student did, so he is subtracting five, then
dividing by 3. These two students are “inverses” of each other—they
will always reverse what the other one does.
e 2Give a few easy functions where people can figure out the inverse.
The inverse of x + 2 is x — 2 (and vice-versa: it is always
symmetrical). The inverse of «° is ¢/x
The key thing to stress is how you test an inverse function. You try a
number. For instance...
10 > 2#+2—7>512->2-2-— 10
—§ > 2@#+2>4-3>2-2->-5
The point is that you take any number and put it into the first function; put
the answer in the second function, and you should get back to your original
number. Testing inverses in this way is more important than finding them,
because it shows that you know what an inverse function means.
¢ 3Ask for the inverse of x. Trick question: it doesn’t have any! Why
not? Because x? turns 3 into 9, and it also turns —3 into 9. It’s allowed
to do that, it’s still a function. But an inverse would therefore have to
turn 9 into both 3 and —3, and a function is not allowed to do that—
rule of consistency. So x? is a function with no inverse. See if the class
can come up with others. (Some include y = |x| and y = 3.)
e 4Now ask them for the inverse of 10 — x. They will guess 10 + 2 or
x — 10; make sure they test! They have to discover for themselves that
these don’t work. The answer is 10 — 2; it is its own inverse. (It turns
7 into 3, and 3 into 7.) Ask for other functions that are their own
inverses, see if they can think of any. (Other examples include y = z,
y = —2, y = 20/z.)
e 5In practice, inverse functions are used to go backwards, as you might
expect. If we have a function that tells us “If you work this many
hours, you will get this much money,” the inverse function tells us “If
you want to make this much money, you have to work this many
hours.” It reverses the x and the y, the dependent and independent
variables. Have the class come up with a couple of examples.
e 6Formally, an inverse function is written f~! (a). This does not mean
it is an exponent, it is just the way you write “inverse function.” The
strict definition is that f ( | ie (a)) = x. This definition utilizes a
composite function! It says that if z goes into the inverse function, and
then the original function, what comes out is...a. This is a hard
concept that requires some talking through.
OK, at this point, you have them start working on the in-class exercise
“Inverse Functions.” Note that you have not yet given them a way of
finding inverse functions, except by noodling around! Let them noodle.
Even for something like y = so -. they should be able to get there, with a
bit of hand-holding, by reversing the steps: first multiply by 7, then subtract
3, then divide by 2. If they ask about #11, make sure they try a few things
(such as ¢/zx) and test them—they will discover they don’t work. Explain
that, in fact, we have no inverse function of 2” right now, so we’re going to
make one up later in the year and call it a “logarithm.” They can then leave
this one blank.
After they have finished noodling their way through the most of the
exercises, interrupt the class and say “Now, I’m going to give you a formal
method of finding inverse functions—you will need this for the
homework.” The formal way is: first, reverse the x and the y, then solve for
. For instance, from y = 22+ we first write 7 = 2y*3 Then solve for
y y = = y
_ Ta—2
to gety= =.
Homework:
“Homework: Inverse Functions”
But wait! We’re not done with this topic!
What happens the next day is, they come in with questions. Whatever else
they did or didn’t get, they got stuck on #10. (If they got stuck on #9, point
out that it is the same as #2; make sure they understand why.) #10 is hard
because they cannot figure out how to solve for y. This brings us, not to a
new conceptual point, but to a very important algebraic trick, which we are
going to learn by doing a TAPPS exercise.
TAPPS (Thinking Aloud Pair Problem Solving) is a powerful learning tool,
and here’s how it goes. The students are broken into pairs.
One person in each pair is the teacher. His job is to walk through the
following solution, step by step, explaining it. For each step, explain two
things. Why am I allowed to do that, and why did I want to do that? Your
explanation should make perfect sense to a normal Algebra I student. You
should never skip steps—go line, by line, by line, explaining each one.
The other person is the student. He also has two jobs. First, whenever the
teacher says something that is not perfectly clear, stop him! Even if you
understand it, say “Wait, that didn’t make perfect sense.” Keep pushing
until the explanation is completely bullet-proof. Second, keep the teacher
talking. If he pauses to think, say “Keep talking. What are you thinking?”
The teacher should “think out loud” until he comes up with something.
If the students are stuck on a line, they should raise their hands and ask you.
Take the time to carefully explain the process—we will do other TAPPS
exercises. And one more thing—warn them that after everyone has
completed the exercise, you will be calling on individuals to explain tricky
steps. You will not call for volunteers. After they are done, everyone in the
class should be able to answer any question about anything in this
derivation. So you will just pick people and ask them questions like “Why
did I do that?”
After they are done, call on individuals and ask questions like “Why did we
subtract 2xy from both sides?” and “How did we get y(1 — 2a)?” Point out
the general strategy is only two steps: get all the y things on one side, then
pull out the y. Check their answer to the question at the end.
Time for another test!
Once again, there is the sample test—you will probably want to assign it as
a homework, and tell them to do that and also study everything since the
last test. The next day, go over the homework and any questions. Then give
the test.
Congratulations, you’re through with your first unit! If things went
well, you have laid the groundwork for the entire year. Onward and
upward from here!
Introduction
Introduction to Chapter 2 of Felder's Algebra 2 Teacher's Guide.
I like to have this unit early in the year, because it introduces another of my
main themes: some of these problems require you to think. There is
almost no “mechanical” way to solve them. I want to set up that expectation
early.
Of course, this is really two very different topics, and both of them contain
an element of review as well as some new material. But by the time we get
to combining them in problems like , it is new to everybody.
Those problems are harder than you might suppose!
Inequalities
A teacher's guide to inequalities.
This is pretty easy, isn’t it? You can just start by having them work through
“Inequalities” with no preamble at all.
After they’ve worked on it for a while, you might want to interrupt the class
to talk about it for a while. #1 is obviously an attempt to get at the old “you
have to reverse the inequality when you multiply or divide by a negative
number” thing. But stress as loudly as you can, the idea that they shouldn’t
just take anyone’s word for it. The question we’re trying to get at is, why
do you have to switch the inequality, then and only then? I usually illustrate
this point by drawing a number line and showing how, on the left of the
zero (in “negative land”) the numbers are going backward (an observation
that every second grader notices, but that they have forgotten by high
school). So you can visually show how 2 < 3 becomes —2 > —3 when you
move it to the other side.
Also, tell them how important it is to distinguish carefully between ANDs
and ORs—this will become a major issue later. I have two pet peeves on
this topic.
One pet peeve is people who memorize a facile rule (such as “less-than
problems become AND and greater than problems become OR”) without
having the slightest idea what they are doing. I go out of my way to create
problems that frustrate such rules (which isn’t hard to do). They have to see
and understand what these conjunctions mean.
The other pet peeve is z > +4. This is, for all intents and purposes,
meaningless. I want them to realize that on #9, and then I warn them that I
will always take off points if they answer any question this way. (This
comes up in the context of || > 4 and is a good example of how wrong
you go if you answer mechanically instead of thinking.) x can equal +4, if
you know what that means (shorthand for z = 4 or x = —4), but it cannot
be greater than, or less than, +anything.
Homework:
“Homework: Inequalities”
Inequality Word Problems
A teacher's guide to inequality word problems.
The difficulty with this is that there is really nothing to say about it at all:
it’s just something I want them to see. So there is this homework
assignment, which you can give at any time, before or after anything else.
Stick it in when you find yourself, at the end of a day, not quite ready to
give out the next homework, or something like that.
Homework:
“Inequality Word Problems.”
In one problem, they have to make up an inequality word problem. You
might think that, just because they have made up so many word problems
and functions by now, they would knock this one down easily—but it ain’t
so. I’ve had students who could create functions all day long (by this time)
who could not create a good inequality to save their lives. They create
scenarios like “I eat three bowls of cereal a day, how many do I eat in a
week?” There is nothing unequal there. Just like everything else, this takes
practice. But I do want them to see that inequality relationships are all
around us.
Absolute Value Equations
A teacher's guide to absolute value equations.
Now it gets tough. But once again, little or no preamble is needed: just have
them start working on “Absolute Value Equations.”
Here’s the thing. Problems 1—9 really contain all the math: all the concepts
they need to get. And, for the most part, they will get them right—although
you need to check this before they go on any further.
But when it comes to the more complicated-looking problems in the second
half of the assignment, they panic. They stop thinking, revert to rules, and
start getting wrong answers. If they are diligently checking, they will realize
that their answer to #12 doesn’t work. But they may need you to point out
that this is because it is analogous to #6 and has no answer.
So, around this time, I spend a lot of time insisting “Think, think, think!”
The way to think it through is this. Once you have solved for the absolute
value, go back to the kind of thinking you did in the first page. For instance,
when you have |x + 3] = —1, cover up the x + 3 and ask yourself the
question like this: “The absolute value of something is —1. What is the
something?” The answer, of course, is “nothing.” Think, and you will get it
right. Plug and chug, and you will get it wrong.
OK, if |x + 3] = 7 has two answers, and |x + 3| = 0 has one, and
|x + 3] = —4 has none, then what about |a—2| = 22-10? The answer is,
you don’t know until you try. You begin by splitting it the same way you
did before: e— 2 = 22-10 or x—-2 =— (22-10). Find both answers. But
then check them: even if you did the math right, they may not work!
Don’t tell them this up front, but make sure to discuss this with them toward
the end of class, in the context of #13; they will need to know this for the
homework.
Homework:
“Homework: Absolute Value Equations”
Absolute Value Inequalities
A teacher's guide to absolute value inequalities.
They are going to work on the assignment “Absolute Value Inequalities” in
class. You may want to begin by reminding them that they have already
been solving absolute value inequalities. On the previous assignment they
turned “the absolute value of my number is less than 7” into an inequality
and solved it, by trying a bunch of numbers. These are no different. We are
going to use the same sort of thinking process as before: confronted with
|3az — 1| > 10 we will say “OK, the absolute value of something is greater
than 10. What could the something be? <think, think> OK, the something
must be greater than 10 (like 11,12,13) or else less than —10 (like —
11,-12,-13).” So we write 3x — 1 > 10 or 32 — 1 < —10 and go from
there.
"Gee, why are you making it so hard? My Algebra I teacher taught me that
if it’s greater than, just make it an “or” and if it’s less than, just make it an
“ang
OK, let’s try a slight variation: |3a — 1] > —10. Now what? “OK, the
absolute value of something is greater than —10. What could the something
be? <think, think> OK, the something can be...anything!” The absolute
value of anything is greater than —10, so any x-value will work!
"That’s not what my Algebra I teacher taught me."
Fine, then, let’s try that same problem your way. Then, let’s test our
answers, by plugging into the original inequality and see which answer
works.
You get the idea? The whole point of this unit (to me) is to—very early in
the year—establish a pattern that the only way to solve math problems is by
thinking about them. This unit is great for that.
Homework:
“Homework: Absolute Value Inequalities”
Graphing Inequalities and Absolute Values
A teacher's guide to graphing inequalities and absolute values.
This is a two-day topic, possibly three.
Start by putting the function y = x?-1 on the board. Now, distinguish
between two different kinds of questions.
1. Solve (or graph the solution of) z2- 1 < 0. This should remind the
students of problems we did in the last unit, where we asked “For what
x-values is this function negative?” The answer is —1 < x < ]; it
could be graphed on a number line.
2. Graph y < «2-1. This is a completely different sort of question: it is
not asking “For what x-values is this true?” It is asking “For what (z,
y) pairs is this true?” The answer cannot live on a number line: it must
be a shaded region on a two-dimensional graph. Which region? Well,
for every point on this curve, the y-value is equal to z2— 1. So if you
go up from there, then y is greater than...but if you go down from
there, then y is less than... So you shade below it.
It is important to be able to solve both types of problems, but it is even
more important, I think, to distinguish between them. If you answer the first
type of question with a shaded area, or the second type on a number line,
then you aren’t just wrong—you’re farther than wrong—you’ re not even
thinking about what the question is asking. (“2 + 2 = 5” is wrong, but “
2 +2 = George Washington” is worse.)
With that behind you, get them started on the assignment “Graphing
Inequalities and Absolute Values.” They should get mostly or entirely
finished in class, and they can finish it up and also do the homework that
evening.
Homework:
“Homework: Graphing Inequalities and Absolute Values”
If they come in the next day asking about #4, by the way, just tell them to
turn it into y < —2|a| and then it is basically like the other ones.
Second day, no worksheet. After going over the homework, stressing the
ways we permute graphs, warn them that here comes a problem that they
cannot solve by permuting. Challenge them—first person with the correct
shape is the winner, no calculators allowed—and then put on the board
y = x + |z|. Give them a couple of minutes to plot points. Then let
someone who got it right put it on the board—both the points, and the
resultant shape.
Now, you point out that this shape is really a combination of two different
lines: y = 2z on the right, and y = 0 on the left. This odd two-part shape is
predictable, without plotting points, if you understand absolute values in a
different way. This is our lead-in to the piecewise definition of the
absolute value:
_f 2 xz > 0
— l-2 «<0
This takes a whole lot of explaining: it is just one of those things that
students find difficult. Here are a few ways to explain it (use all of them).
¢ iJust try numbers. If = 3, then |x| = 3, so |x| = x. Same for x = 4,
x = 52, and even x = 0. But if x = —3, then |x| = 3, so |x| A x
(they are not the same)! Instead, |2| = —a. Why, because — x in this
case is —(—3) which is 3 which is indeed |].
But how can |z| = —z when |z| is never negative? Well, that brings us
1Ops
e 2Putting a— sign in front of a number does not make it negative: it
switches the sign. It makes positive numbers negative, and negative
numbers positive. So you can read that piecewise definition as “if x is
negative, then the absolute value switches the sign.”
e 3Finally, come back to the graph of good old y = |z]|. Point out that it
is, indeed, the graph of y = z on the right, and the graph of y = —x
on the left.
Now, how does all this relate to our original problem? When z < 0, we
replace |x| with —z so our function becomes y = x — x = 0. When z > 0,
we replace |x| with x so our function becomes y = x + x = 2a. That’s
why the graph came out the way it did.
Why is this important? It’s an important way to understand what absolute
value means. But it’s also our first look at piecewise functions (one of the
only looks we will get) so take a brief timeout to talk about why piecewise
functions are so important. Throw an object into the air and let it drop, and
talk about the function h(t). We previously discussed this function only
during the flight. But to get more general, you have to break it into three
different functions: h = 3 before you throw it (assuming it was in your
hand 3' above the ground), h = 16 — t? or something like that during the
flight, and h = 0 after it hits the ground. Do a few more examples to get the
idea across that piecewise functions come up all the time because
conditions change all the time.
OK, back to our friend the absolute value. The students should now graph
y= cy on their own (individually, not in groups), not by plotting points,
but by breaking it down into three regions: x < 0, x = 0, andz > 0. (It is
different in all three.) Get the right graph on the board.
Now, hopefully, you have at least 10-15 minutes left of class, because now
comes the hardest thing of all. You’re going to graph |x| + |y| = 4. Since x
is under the absolute value, we have to break it into two pieces—the left
and the right—just as we have been doing. Since y is under the absolute
value, we also have to break it vertically. So what we wind up doing is
looking at each quadrant separately. For instance, in the second quadrant,
x < 0 (so we replace |x| with —«) and y > 0 (so we replace |y| with y). So
we have y — x = 4 which we then put into y = mz + b format and graph,
but only in the second quadrant. You do all four quadrants separately.
Explain this whole process—how to divide it up into the four quadrants,
and how to rewrite the equation in the second quadrant. Then, set them
going in groups to work the problem. Walk around and help. By the end of
the class, most of them should have a diamond shape.
After they are all done, you may want to mention to them that this exact
problem is worked out in the “Conceptual Explanations” at the very end of
this chapter. So they can see it again, with explanations.
Homework:
Graph |x| — 2|y| < 4. This requires looking at each quadrant as a separate
inequality and graphing them all in the appropriate places!
Time for another test!
Once again, there is the sample test—you will probably want to assign it as
a homework, and tell them to do that and also study everything since the
last test. The next day, go over the homework and any questions. Then give
the test.
Introduction to Simultaneous Equations
Like all our topics so far, this unit reviews something the students covered
in Algebra I—but it goes deeper.
Begin by having them work their way through the assignment “Distance,
Rate, and Time” in pairs. Most of it should be pretty easy, including getting
to the general relation d = rt. It should not take much time.
Until the last question, that is. Let them work on this for a while. Some will
get all the way, some will not get very far at all. But after they’ve been at it
for a while, tell them to stop working and pull them back to a classwide
discussion. Show them how to set up d = rt for each case, and make sure
they understand. Maybe come up with another problem or two along the
same lines, including one where the times are the same and the distances
are different. (A train leaves Chicago and a train leaves New York, when do
they crash...) Get them comfortable with setting up the two equations—
we're not really focused on solving them.
Toward the end, see if they remember that there are three ways of solving
these equations. Two of them, Substitution and Elimination, will be covered
tomorrow. Tonight, on homework, we are going to solve by graphing. Your
big job is to drive home the point that since a graph represents all the points
where a particular relationship is true, therefore the place where the two
graphs intersect is the point where both relationships are true. Also talk
about the fact that graphing is not 100% accurate (you sort of eyeball a
point and say “it looks like around this”) and how to check your answer
(plug it back into both equations).
Homework:
“Homework: Simultaneous Equations by Graphing”
Simultaneous Equations
At the start of the second day, explain the two techniques of substitution
and elimination. This is review, so they should get it with just a few
examples.
Then have them do the assignment “Simultaneous Equations.”
At the end of class, if they are mostly done, they can finish that for
homework and also do the homework. If they are not mostly done, they can
just finish it for homework, and you can have them do the “homework” the
next day in class.
Homework:
“Homework: Simultaneous Equations”
The "Generic" Simultaneous Equation
This should be done in class as another TAPPS exercise. Remind them of
the ground rules, and especially of the fact that you will be asking them
questions afterwards to make sure they got it.
Time for another test!
Once again, there is the sample test. If everyone finishes the TAPPS
exercise early, and they all seem pretty comfortable with the material, you
may not want to do the “do the sample test tonight and we’ Il go over it
tomorrow and then have the test the next day”—tomorrow may be pretty
boring! Instead, it may be OK (depending on the class) to just say “Now
work in class on the sample test, use it to help you study tonight, and we
will have a real test tomorrow.”
Introduction
There are really three separate pieces of this unit: factoring, solving
quadratic equations, and graphing quadratic functions. The first piece is
vital and important, but small. Nonetheless, you may want to add a small
quiz between that section and the next. I have included two sample tests—
the first on factoring and solving quadratic equations, the second on
graphing.
Multiplying Binomials
Sounds trivial, doesn’t it? But this is one of the most important days in the year.
What they do know, from Algebra I, is how to FOIL. This takes two seconds of review and you’re done.
However, there are two points that their Algebra I teacher never made.
1. When we say (x + 3) (x + 4) = x? + 7x + 12, we are asserting the equality of two functions—that
is, if I plug any number into (xz + 3)(x + 4), and plug that same number into x? + 7x + 12, it should
come out the same. It’s an algebraic generalization.
2. FOIL leaves you high and dry if you have to multiply (2 + 2)(a + y+ 3). The real algorithm for
multiplying polynomials is to multiply everything on the left by everything on the right. Walk
through an example of this on the board. Show them how FOIL is just a special case of this rule, with
both things are binomials.
At this point, they can start working in pairs on the exercise “Multiplying Binomials.” They should have no
problem with the first few. As you are walking around, your main job is to make sure that they are doing #5
correctly. They should not be multiplying these out explicitly (so that (2 + 4)(a + 4) becomes
x? + 4a + do + 16) and then combining the middle terms. They should instead be using the formula that
they just developed, (x + a)? = x? + 2ax + a? to jump straight to the right answer. A lot of them will find
this very confusing. I always explain it this way: x and a are both placeholders that could represent
anything. So when we say:
(w +a)? =a? + 2ax + a?
what we’re really saying is:
(something + something else)” = something” + 2(something) (something _else) + something else”
Maybe walk them through the first one as an example.
One point of this exercise is to get them to the point where they can see immediately, with no in-between
steps, that (@ + 4)? = 27+ 8x + 16. Some of them will think that this new, confusing method may be
faster, but they can just go right on doing it the “old way” with FOIL. I always explain to them that, in a few
days, we’ll be learning a technique called completing the square that involves reversing this formula, and
therefore cannot possibly be done with FOIL. They need to know the formula.
Another point is to get our three formulae on the table: (2 + a)?, (x — a), and x? — a?. There are very few
things I ask the class to memorize during the year, but these three formulae should all be committed to
memory.
But the larger point is to give them a new understanding and appreciation for what variables do—to
understand that x and a represent anything, so that once you have a formula for (x + a)? you can use that
formula directly to find (2y + 6z)?.
Homework:
“Homework: Multiplying Binomials”
Factoring
Begin class by reminding them of what they already know: factoring means
turning x? + 7x + 12 into (x + 3)(a + 4). Then ask—how would we
check that? There are two ways. First, we can multiply it back (using FOIL
for instance). Second, we can try a number (since we are making the claim
that these two functions are “equal,” remember?). Stress this very heavily:
they have to know both ways of checking. Why? Because if you don’t
know both ways of checking, then you don’t really understand what
factoring is, even if you get the right answer.
OK, on to...how do you do it? There are three steps to factoring.
1. Pull out common terms. This is always the first step!
2. Use the formulae from yesterday. For instance, given x? — 9, you
recognize it as the difference between two squares. Given x? — 6x + 9
, you recognize it as (x — 3)?.
3. When all else fails, plain old-fashioned factoring. Do a few examples
on the board, just to refresh their memory. x? + 7x +12,
x? — Te +12, 2? + 2 — 12, x? — x — 12 are good starting examples,
and give you an opportunity to talk about what effect negative
numbers have, both in the middle term (not much) and in the last term
(lots). The thing I always stress is to start with the last term—find all
the pairs of numbers that multiply to give you the last term, and then
see if any of them add to give you the middle term. The real test is
when you are faced with something like x? + 4x + 8; it should not
take long to determine that it cannot be factored!
Now they are ready to start on the “Factoring” assignment during class.
Homework:
“Homewoik: Factah Alla Dese Heah Spressions”
As I mentioned, this might be a good place to break and have a quiz. Or it
might not. What do I know? Anyway, on to quadratics...
Introduction to Quadratic Equations
Get them started on the assignment “Introduction to Quadratic Equations”
with little or no preamble. Then, after a few minutes—after everyone has
gotten through #5—stop them.
Make sure they all got the right answers to numbers 2 and 3, and that they
understand them. If zy = 0 then either z, or y, must equal zero. There is
no other way for it to happen. On the other hand, if cy = 1, that doesn’t tell
you much—either one of them could be anything (except zero).
Now, show how this relates to quadratic equations. They do remember how
to solve quadratic equations by factoring. 2? + e—12 = 0,
(x + 4)(x — 3) = 0, x = —4 rz = 3. But that last step is taken as a
random leap, “because they told me so.” The thing I want them to realize is
that, when they write (xz + 4)(a — 3) = 0, they are in fact asserting that
“these two numbers multiply to give zero,” so one of them has to be zero.
This helps reinforce the idea of the previous lesson, that x and y can mean
anything: (x + 4)(x — 3) = O is in fact a special case of ry = 0.
The acid test is, what do you do with (x + 5)(a + 3) = 3? The ones who
don’t get it will turn it into x + 5 = 3,2 +3 = 3. And get two wrong
answers. Instead you have to multiply it out, then get everything on one
side so the other side is 0, and then factor.
Now they can keep going. Many of them will need help with #6—talk them
through it if they need help, but make as much of it as possible come from
them. This is a very standard sort of “why we need quadratic equations”
type of problem.
The last four problems are a sneaky glimpse ahead at completing the
square. For #11, many students will say x = 3; remind them that it can also
be —3. For #12, this is yet another good example of the “az can be anything”
rule, and should remind them in some ways of the work we did with
absolute values: something? = 9, so something = +3. #13 is obviously
#12 rewritten, and #14 can be turned into #13 by adding 16 to both sides.
Homework:
“Homework: Introduction to Quadratic Equations.”
Completing the Square
When you’re going over the homework, talk for a while about the throwing-
a-ball-into-the-air scenario. It will come up again, and I really want people
to understand it. The particular point I try to make is how the math reflects
the reality. You have a function h(t) where if you plug in any ¢ at all, you
will get an h. You’re using it backward, specifying h and asking for ¢ (as in,
“when will the ball hit the ground?”). What kind of answers would you
expect? Well, suppose you throw the ball 16 ft in the air. If you ask “When
will it be at 20ft?” you would expect to get no answer at all. If you ask
“When will it be at 5 ft?” you would expect two answers—one on the way
up, and one on the way down. If you ask “When will it be at 16 ft?” you
would expect exactly one answer. In all three cases, the math gives you
exactly what you expect.
On the other hand, suppose you ask “When will it be at —3 ft?” (That is,
under the ground.) You might expect no answer at all, since the ball never is
under the ground. But the math doesn’t know that—it thinks the ball is
following the same function forever. So you get two answers. One is after
the ball hits the ground. The other is before it left—a negative time! This is
where you have to use common sense to find the “real” answer, as distinct
from the answer the math gave you.
I spend a good half-period, at least, talking through this. I think it is an
incredibly important point about the way we use math to model the world.
See this webpage for an exercise you can use just on this.
Anyway, onward...the assignment “Completing the Square” pretty much
speaks for itself. Probably the only preamble you need is to point out that
many quadratic equations, which do have solutions, cannot be factored. So
we are going to learn another technique which has the advantage that it can
always be used. (Factoring is still easier and faster when it works.)
Now you can just get them started on it, and then wander around and help.
Just make sure that before the class is done, everyone gets the technique.
You may also want to point out to them that they already did this on
yesterday’s assignment.
On #4 make sure they get two answers, not just one!
Homework:
“Homework: Completing the Square”. The hard ones here, that you will get
questions on the next day, are #9 and #10. Note that, on #9, I am not
looking for the discriminant and the quadratic formula and stuff; just the
obvious fact, based on completing the square, that if c < 0 we have no real
answers, if c = 0 we have one, and if c > 0 we have two. #10 is worth
looking at closely if there are questions, because it leads to the next day.
The "Generic" Quadratic Equation
Begin by reminding them of what we did with simultaneous equations.
First, we learned how to solve them (using substitution or elimination).
Then we used those exact same techniques to solve the generic version—
that is, simultaneous equations where all the numbers were replaced by
letters. This, in turn, gave us a formula that could instantly be used to solve
any pair of simultaneous equations.
Now we are going to do that same thing with quadratic equations. The
“generic” quadratic equation is, of course, ax? + bz + c = 0. Now, we
have learned two different ways of solving such equations. The “generic”
version is hard to solve by factoring (although it is possible); we are going
to do it by completing the square.
Make sure they look over my example of completing the square; this might
be a good opportunity for a quick TAPPS exercise. There are other
examples in the “Conceptual Explanations” so you could do two TAPPS
exercises—that way everyone gets a chance to be the teacher.
Then have them work through the sheet. They should derive the quadratic
formula, and then use it.
By the time they are done, they should have two things. They should have
the quadratic formula, and a bit of practice using it—so now we have three
different techniques for solving quadratic equations. They should also have
derived the formula. I always warn them that I will ask for this derivation
on the next test: it is not enough to know the formula (although that too is
good), you have to be able to derive it.
At the end of class, you may want to talk for just a couple of minutes about
the discriminant, in reference to #11. It should be fairly obvious by that
point to most of them.
Homework:
“Homework: Solving Quadratic Equations”
Time for another test!
As always, there is the sample test, which may or may not be assigned as a
homework. Then there is the test—on multiplying polynomials, on
factoring, and mostly on solving quadratic equations. Make it shorter than
my sample ©
Graphing Quadratic Functions
OK, we’re done solving quadratic equations—we already have three
techniques and that’s enough. But—one thing I say a million times
throughout my class—you never really understand a function until you
graph it.
So, they can do the exercise “Graphing Quadratic Functions.” It doesn’t
require any buildup, they can do it right now. Note that we are not
introducing any of the formal machinery of parabolas (focus, directrix, etc.)
—all that will come much later, in the unit on conics. We are graphing both
horizontal and vertical parabolas the way we did in the very first unit on
functions—by taking an initial starting point (y = x? or z = y’) and
moving it up and down and left and right and stretching it and turning it
upside-down. None of this should require a calculator.
It might be worth mentioning that a horizontal parabola is not a function.
But we can still talk about it and graph it.
Homework:
Finish the in-class assignment. That was a long one, wasn’t it?
Graphing Quadratic Functions IT
The beginning of the “Graphing Quadratic Functions II” exercise is review
of yesterday. After letting them work on it together, you may want to
interrupt and have them do the thing in the middle as a TAPPS exercise.
The key things you need to ask them about are how this is the same as, and
different from, the way we completed the square before. For instance, we
used to add nine to both sides (because, let’s face it, we had two sides).
Now we only have one side, so we add 9 to it, and subtract 9 from it, at the
same time. This gives us what we want (the perfect square) without
changing the function.
Homework:
“Homework: Graphing Quadratic Functions II.” You will get questions the
next day about #8 (which is really a line) and #11 (which they just flat can’t
graph at this point). These lead nicely into #12. If it has an x? but no y’, it’s
a vertical parabola. If it has a y* but no 2”, it’s a horizontal parabola.
Solving Problems by Graphing Quadratic Functions
Now, at long last, we see a use for all this graphing we’ve been doing.
In the throwing-a-ball scenario, we have an h(t) that can be used to answer
two kinds of questions. “I know the time, but what is the height?” (easy,
plug in) and “I know the height, what is the time?” (harder, requires solving
a quadratic equation). But there is a third kind of question, very important
in the real world, which is: “How high does it go?” Now we don’t know the
time or the height! But if we graph it, and find the vertex, we can find both.
Now they can work a while on the in-class assignment. Many of them will
get stuck dead on #3. This is where you have to pull back and lecture a bit
more. Help them draw it, and set up the function A(x). But more
importantly, talk about what that function means. You plug in any x
(length) and you get back an A (area). So, if the graph looks like this n
what does that tell us? Well, at the peak there, that is the highest A ever gets
on our graph—that is, the highest the area ever gets. Find the vertex, and
you will find the x that maximizes A!
This is worth a lot of time to make sure people really get it. It comes all the
way back to week 1, and the idea of graphing a function. On one level, it’s
incredibly abstract—we are drawing an upside-down parabola that
somehow represents the “possibility spaces” for a bunch of rectangles. But
if you understand the idea of graphing a function, it is really very simple.
Every point on that parabola pairs an x (length) with an A (area). Every
point represents one farm that our farmer could create. It’s obvious, looking
at it, that this point at the top here represents the one with the highest area.
This is one of those cases where the in-class assignment and the homework,
together, could easily take two days instead of one. Let it take that, if it
does. Make up more problems, if you have to. But don’t let them get away
with thinking “I understand everything else, I just don’t get the word
problems.”
Homework:
“Homework: Solving Problems by Graphing Quadratic Functions”
Quadratic Inequalities
For some reason, this is one of the hardest topics in the course. It shouldn’t
be hard. There is nothing hard about it. But students get incredibly tied in
knots on this, by trying to take short cuts. The hardest part is convincing
them that they have to think about it graphically.
So, begin by simply putting these two problems on the board.
x?— 32-4 > 0
a’-32+3>0
Allow them to work in pairs or groups. Offer a piece of candy or a bit of
extra credit or some such to anyone who can find the answer to both
problems. Give them time to really work it. Almost no one will get it right,
and that’s the point. It’s very hard to think about a problem like this
algebraically. It’s very easy if you think about it the right way: by graphing.
So, we’re going to graph both of those functions. But strangely enough,
we’re going to do it without completing the square or finding the vertex. In
each case, we’re only going to ask two questions: what are the zeros of the
function, and which direction does it open in? These two questions are all
we need to answer the inequality.
In the first case, by factoring, we find two zeros: 4, and —1. In the second
case, we find with the quadratic formula that there are no zeros. Both
graphs open up. (Why? Because the coefficient of the x” term is positive.)
So the graphs look something like this.
y
What are the vertices, exactly? We don’t know. If we wanted to know that,
we would have to complete the square, just as we did before.
But if all we want to know is where each graph is positive, we now have
it. The graph of the first function should make it clear that all numbers to
the right of 4 work, as do the numbers to the left of —1, but the numbers in
between don’t work. (Quick review: how can we write that answer with
inequalities? With set notation?) The graph of the second function makes it
clear that all numbers work.
Check this by trying numbers in the original inequalities.
Now, challenge them to find a quadratic inequality that is in the form
f(x) > 0 where f(a) is a quadratic function and the solution is nothing
works. The key is, of course, it has to be an upside-down quadratic.
Homework:
“Homework: Quadratic Inequalities”
Time for another test II!
Congratulations, kids! We are done with our entire unit on quadratic
equations—probably the biggest unit in the course. (Certainly the only think
I remember from my own Algebra II course.)
Introduction
A introduction to the teacher's guide on exponents.
After all the incredibly new stuff we’ve been doing, it’s a nice break to get
back to something with a large element of review in it.
But it is also a problem. Many of the kids know already that 232° = 28. A
fair number of them even know that 2~’ = 1/27. But they don’t know
why. It’s vital to keep reminding them that it isn’t enough to know it, they
have to know why—and this will indeed be reflected on the test.
Rules of Exponents
A teacher's guide to the rules of exponents.
Yes, a lot of this assignment was already done, verbatim, in the unit on
Functions. But there are a lot of reasons for bringing it back (just as we did
in Quadratics). First, they (and you) can discover that they have gotten
better at finding generalizations. But more importantly, back when we did
this in functions, we were only interested in the process of finding
generalizations. Now we are focused on creating, memorizing, and using
the three rules of exponents.
2d is really pushing them another step toward “the way mathematicians
think”—seeing that 2 (27) — 27+! is really just a special case of
279 — 22+ wherea = 1.
In #3, I really want to get at the idea that 2 so = — 37, and — = on In other
words, the whole thing can and should be done without negative exponents.
Why? Because we haven’t yet defined what they mean—and why.
Homework:
“Homework: Rules of Exponents”
Extending the Idea of Exponents
A teacher's guide to extending the concept of exponents to students.
This is one of my favorite class discussions. You’re going to talk almost the
entire class, and just give out an assignment toward the end.
So far, we have only talked about exponents in the context of positive
integers. The base can be anything: for instance, we can find (—3)* or ($) =
. But when we say that 2° means 2 e 2 e 2, that definition is really only
meaningful if the exponent is a positive integer. We can’t multiply 2 by
itself “ —3 times” or «= times.” Or “O times” for that matter.
So, let’s plop ourselves down in an imaginary point in history where
exponents are only defined for positive integers. We are the king’s
mathematicians. The king has just walked in and demanded that we come
up with some sort of definition for what 2~* means. “Zero and negative
numbers have rights too,” he growls. “They must be treated equally, and
given equal rights to be in the exponent.”
So we start with a brainstorming exercise. The object is to come up with as
many possible things as you can think of, for 2~? to mean. As always, this
should be done in groups of 2 or 3, and remind them that the object is
quantity, not quality: let’s get creative. If you can’t think of more than two
or three definitions, you’re not trying hard enough.
By the way, somewhere in class there is a smart-aleck who knows the right
answer and therefore won’t plan. “It’s 1/2” he insists proudly. “Why are
we doing this?” To which you reply: “The people in the class are coming up
with dozens of things it could mean. Can you give them a good argument as
to why it should mean that, instead of all the others?” And he weakly
answers “Well, my Algebra I teacher told me...” and you’ve won. The
point, you explain, is not to parrot what your teacher told you, but to
understand several things. The first is that our old definition of an exponent
(“multiply by itself this many times”) just doesn’t apply here, and so we
need quite literally a new definition. The second is that there are a ton of
definitions we could choose, and it frankly seems arbitrary which one we
pick. The third is that there really is a good reason for choosing one and
only one definition. But if he doesn’t know what that he, how about if he
gets with the program and brainstorms? End of discussion.
OK, so after a few minutes, you start collecting ideas from the groups. 2~°
means... 2°, only negative (so it’s —8). It means 2 divided by itself three
times, 2/2/2 (so it’s either 2 or + depending on how you parenthesize it).
And so on. With a bunch of ideas on the board, you say, now we have to
choose one. How do we do that? That is, what criteria do we use to decide
that one definition is better than the others? (silence)
The answer is—the definition should be as consistent as possible with the
one we already have. Of course it won’t mean the same thing. But it should
behave mathematically consistently with the rules we have: for instance, it
should still obey our three laws of exponents. That sort of consistency is
going to be the guideline that we use to choose a definition for the king.
And hey, what exactly do negative numbers mean anyway? One way to
look at them is, they are what happens when you take the positive numbers
and keep going down. That is, if you go from 5 to 4 to 3 and just keep
going, you eventually get to 0 and then negative numbers. This alone is a
very powerful way of looking at negative numbers. You can use this to see,
for instance, why positive-times-negative-equals-negative and why
negative-times-negative-equals-positive.
3e5=15 3e—5=-—15
20e5= 10 2e—5= —10
le5=5 le—5=-—5
0e5=0 0e—-5=0
—le5=—5 —le-5=5
—2e5=—10 —2e—5= 10
OK, you may or may not want to get into this, but I think it’s cool, and it
does help pave the way for where we’re going. (It also helps reinforce the
idea that even the rules you learned in second grade have reasons.) The
numbers I’ve written on the left there show what happens to “multiply-by-
5” as you count down. Clearly, looking at the positive numbers, the answers
are going down by 5 every time. So if that trend continues as we dip into
negative numbers, then we will get —5 and then —10 on the bottom: negative
times positive equals negative.
On the right, we see what happens to “multiply-by--5” as you count down.
Since we already know (just proved) that positive-times-negative-equals-
negative, we know that 3 e —5 = —15 and so on. But what is happening to
these answers as we count down? They are going up by 5. So, continuing
the trend, we find that —1 e —5 = 5 andso on.
As I say, you may want to skip that. What is essential is to get across the
point that we need a new definition that will cover negative exponents, and
that we are going to get there by looking for consistency with the positive
ones. Then they are ready for the in-class assignment: it shouldn’t take long.
Do make sure to give them 10 minutes for it, though—you want them to
finish it in class, and have time to ask you questions, so you know they are
ready for the homework.
Homework:
“Homework: Extending the Idea of Exponents”
Fractional Exponents
A teacher's guide to fractional exponents.
Start by reminding them of where we are, in the big picture. We started with
nothing but the idea that exponents mean “multiply by itself a bunch of
times”—in other words, 74 means 7 e 7 e 7 e 7. We went from there to the
rules of exponents— x?x° = x%+ and so on—by common sense. Then we
said, OK, our definition only works if the exponent is a positive integer. So
we found new definitions for zero and negative exponents, but extending
down from the positive ones.
Now, we don’t have a definition for fractional exponents. Just as with
negative numbers, there are lots of definitions we could make up, but we
want to choose one carefully. And we can’t get there using the same trick
we used before (you can’t just count and “keep going” and end up at the
fractions). But we still have our rules of exponents. So we’re going to see
what sort of definition of fractional exponents allows us to keep our rules of
exponents.
From there, you just let them start working. I can summarize everything on
the assignment in two lines.
1. The rules of exponents say that («*) 2 — x. So whatever x? is, we
know that when we square it, we get x. Which means, by definition,
1 =
that it must be «/z. Similarly, 23 = ¥/z and so on.
1 2
2. The rules of exponents say that (« *) 2 — x3. Since we now know
1 2 2 j
that x = ¥/z, that means that 2? = (¥/zx)”. So there you have it.
Thirty seconds, written that way. A whole class period to try to get the
students to arrive their on their own, and even there, many of them will
require a lot of help to see the point. Toward the end, you may just call the
class’s attention to the board and write out the answers. But by the time they
leave, you want them to have the following rule: for fractional exponents,
the denominator is a root and the numerator is an exponent. And they
should have some sense, at least, that this rule followed from the rules of
exponents.
There is one more thing I really want them to begin to get. If a problem
ends up with / 25, you shouldn’t leave it like that. You should call it 5. But
if a problem ends up with V2, you should leave it like that: don’t type it
into the calculator and round off. This is also worth explicitly mentioning
toward the end.
Homework:
“Homework: Fractional Exponents”. Mostly this is practicing what they
learned in class. The inverse functions are a good exercise: it forces them to
review an old topic, but also forces them to practice the current topic. For
instance, to find the inverse function of y=x*/, you write:
= Vee Sy yav se =e.
After you go through that exercise a few times, you start to see the pattern
that the inverse function actually inverts the exponent. The extra fun comes
when you realize that x° has no inverse function, just as this rule would
predict.
ew] to
tole
r= Y
At the end of the homework, they do some graphs—just by plotting points
—you will want to make sure they got the shapes right, because this paves
the way for the next topic. When going over the homework the next day,
sketch the shapes quickly and point out that, on the graph of 2”, every time
you move on to the right, y doubles. On the graph of (4) *, every time you
move one to the right, y drops in half.
“Real Life” Exponential Curves
A teacher's guide to example exponential curves.
This is another one of those topics where the in-class exercise and the
homework may take a total of two days, combined, instead of just one. This
is a difficult and important topic.
We begin with a lecture something like the following:
Earlier this year, we talked about “linear functions”: they add a certain
amount every time. For instance, if you gain $5 every hour, then the graph
of your money vs. time will be a line: every hour, the total will add 5. The
amount you gain each hour (5 in this case) is the slope.
Can a line also subtract every day? Sure! That isn’t a different rule, because
adding is the same as subtracting a negative number. So if Mr. Felder is
losing ten hairs a day, and you graph his hairs vs. time, the graph will be a
line going down. The total subtracts 10 every day, but another way of
saying that is, it adds —10 every day. The slope is —10. This is still a linear
function.
So why am I telling you all this? Because “exponential functions” are very
similar, except that they multiply by the same thing every time. And, just as
linear functions can subtract (by adding negative numbers), exponential
functions can divide (by multiplying by fractions: for instance, multiplying
by + is the same as dividing by 3). The amount you multiply by is called...
well, come to think of it, it doesn’t have a cool name like “slope.” I guess
we could call it the “base.”
Then they can begin to work on the assignment. They will make it through
the table all right. But when it comes to finding the formula for the nth day,
many will fall down. Here is a way to help them. Go back to the table and
say: “On day 3, let’s not write “4”—even though it is 4 pennies. It is 2 times
the previous amount, so let’s just write that: 2 x 2. On day 4, it’s 2 times
that amount, or 2 x 2 x 2. On day 5, it’s 2 times that amount, or
2x 2 x 2 x 2. This is getting tedious...what’s a shorter way we can write
that?” Once they have expressed every answer in powers of 2, they should
be able to see the 2”~! generalization. If they get the wrong generalization,
step them through to the next paragraph, where they test to see if they got
the right answer for day 30.
You go through the same thing on the compound interest, only harder. A lot
of hand-holding. If you end one year with x then the bank gives you .06z
so you now have a total of z + .06z which is, in fact, 1.062. So, hey, your
money is multiplying by 1.06 every year! Which means if you started with
$1000 then the next year you had $1000 e 1.06. And the next year, you
multiplied that by 1.06, so then you had $1000 x 1.06 x 1.06. And the
year after that....
Toward the end of class, put that formula, $1000 x 1.06n, on the board.
Explain to them that they can read it this way: “Just looking at it, we can
see that it is saying you have $1000 multiplied by 1.06, n times.” This is
always the way to think about exponential functions—you are multiplying
by something a bunch of times.
The assignment is also meant to bring out one other point that you want to
mention explicitly at the end. When we developed our definitions of
negative and fractional exponents, we wanted them to follow the rules of
exponents and so on. But now they are coming up in a much more practical
context, and we have a new need. We want x27 to be bigger than x? and
smaller than x°, right? After all, after 2/4 years, you certainly expect to
have more money than you had at the beginning of the year! It isn’t obvious
at all that our definition, ar =v x°, will have that property: and if it
doesn’t, it’s useless in the real world, even if it makes mathematicians
happy. Fortunately, it does work out exactly that way.
Homework:
“Homework: ‘Real life’ exponential curves”
Time for Another Test!
The sample test will serve as a good reminder of all the topics we’ve
covered here. It will also alert them that knowing why x? is defined the
way it is really does count. And it will give them a bit more practice
(much-needed) with compound interest.
Introduction
The introduction to a teacher's guide on logarithms.
I talk to a surprising number of math teachers who are really uncomfortable
with logs. There’s something about this topic that just makes people
squeamish in Algebra, in the same way that “proving a series converges”
makes people squeamish in Calculus.
It doesn’t have to be hard. It is not intrinsically more complicated than a
radical. When you see ¢/z you are seeing a mathematical question: “What
number, raised to the 3rd power, gives me x?” When you see logsz you are
seeing a question which is quite similar: “3, raised to what power, gives me
x?” I say this about a hundred times a day during this section. My students
may forget the rules of logs and they may forget what a common log is and
they will almost certainly forget e, but none of them will forget that log.8
means the question “2 to what power is 8?” You may want to show them the
“Few Quick Examples” at the beginning of the Conceptual Explanations
chapter to drive the point home.
It is possible to take any arbitrary logarithm on a standard scientific or
graphing calculator. I deliberately never mention this fact to my students,
until the entire unit (including the test) is over. Faced with log,8 I want
them to think it through and realize that the answer is 3 because 2? = 8.
The good news is, none of them will figure out how to do that problem on
the calculator, if you don’t tell them.
Introduction to Logarithms
This is a pretty short, self-explanatory exercise. There isn’t anything you
need to say before it. But you do need to do some talking after the
assignment. Introduce the word “log” and explain it, as I explained it above:
log,8 means “2 to what power is 8?” Also discuss the fact that the log is
always the inverse of the exponential function.
After they have done the assignment, and heard your explanation of the
word log, then they are ready for the homework. It wouldn’t hurt if that
happens in the middle of the class, so they can get started on the homework
in class, and finish it up at home. The in-class exercise is short, the
homework is long.
“Homework: Logs”
When going over the homework the next day, #20 can be explained two
ways. First: 5 to what power is 54? When asked that way, it’s easy, isn’t it?
You don’t have to find what 54 is, to see that the answer is 4! But there is
also another way to explain it, which gets back to the idea of 5® and log; x
being inverse functions. The first function turns 4 into 5*. So the second
one has to reverse this process, and turn 54 back into 4. This way is harder
to understand, but it makes it a lot easier to see why #21 also has to be 4.
Then, there is the graph—as always, make sure they get the right general
shape. Point out that the most salient feature of this graph is that it grows...
incredibly...slowly as you go farther out to the right. (Every time x
doubles, the graph just goes up by 1.) This is a lot of what makes logs
useful, as we will see.
Logarithms -- Properties of Logarithms
This is very standard stuff. Using the in-class exercise in groups of 2 or 3,
they should be able to find—in some cases with a bit of help from you—
some rules of logarithms. In the homework, they practice using those rules.
One thing I don’t do in the worksheets is formally prove the rules.
However, I have been known to “throw in” the proofs sometimes in class,
either for a group that finishes early, or for the whole class if enough people
are interested. One of the proofs is provided as an example in the
“Conceptual Explanations” along with guidelines for the other two.
But what I really care about is giving them an intuitive grasp of why the
rules work, rather than the proof. The intuitive grasp is what comes from
the exercise, from realizing that the logarithm is essentially a counter. Once
you see that log,8 is asking how many 2s there are in 8, then it’s obvious
that log,(8 e 16) will add up all the 2s in 8, and in 16.
Homework:
“Homework: Properties of Logarithms”
Logarithms -- Using the Laws of Logarithms
Once you have gone through the laws of logarithms, you can spend five
minutes working a couple of problems on the board, like:
log; (2) = log;(3)
and then
log;(x + 1) + log;(x — 1) = log; (8)
The first establishes that if you have log(iy;.) = log¢hat), then this must
equal that. The second shows how you have to use the laws of logs to get
into that form. (The OK students will answer 3. The better students will
answer +3. Only the very best will get +3 and then realize that the —3 is,
after all, invalid! But all of that is a detail, of course.)
Anyway, then there is the worksheet full of problems like that, which also
gives good review of a number of old topics.
The thing is, this really isn’t a whole day. Sneak it in when you have 15-20
minutes left in class. It doesn’t matter whether it comes before, after, or in
the middle of the next topic.
So What Are Logarithms Good For, Anyway?
A teacher's guide to the applications of logarithmic functions.
As always, things get harder when we get into word problems. There are a
few things I want them to take away here.
First—logs are used in a wide variety of real world situations.
Second—logs are used because they compress scales. In other words,
because they grow so slowly, we use logarithmic scales whenever we want
to work with a function that, by itself, grows too quickly. Or, to put it
another way, we use logarithms whenever something varies so much that
you don’t care exactly what is, just what the power of 10 is. Don’t say all
this before they start working, but hopefully they will come up with
something like this on #6.
Homework:
“Homework: What Are Logarithms Good For, Anyway?”
In addition to following up on the in-class work, the homework here also
introduces the common and natural logs. It’s a bit of a weak connection, but
I had to stick them somewhere.
Time for Another Test!
The sample test is actually pretty important here. It pulls together a lot of
ideas that have been covered pretty quickly.
The extra credit is just a pun. The answer is log cabin or, better yet, natural
log cabin. Who says math can’t be fun?
According to my reckoning, you are now approximately halfway through
the curriculum. Mid-terms are approaching. If there are a couple of weeks
before mid-terms, I would not recommend going on to radicals—spend a
couple of weeks reviewing. Each topic (each test, really) can stand a whole
day of review. It may be the most important time in the whole class!
Rational Expressions -- Introduction
We’ ve talked about the word “rational”—it doesn’t mean “sane,” it means a
“ratio” or, in other words, a fraction. A rational expression is just a fraction
with variables.
This section is unique, perhaps, in the fact that it introduces practically no
new skills. They have to be able to factor; they have to know the rules of
exponents; they have to be able to work with fractions; they even have to be
able to do long division. There is nothing new in any of that. It’s just putting
it all together to simplify, and work with, rational expressions.
Part of the benefit of this unit is that there are always a few kids in class—
maybe more than a few—who have a lingering, secret fraction-phobia.
They are hoping that no one will ever notice because the calculator will
always rescue them. You can spot these people because they always answer
everything—including “what is 2 divided by 3?”—in decimals. But this unit
will flush them out. You can’t get through rational expressions unless you
know how to do fractions, and your calculator will not help you. (I always
point this out, very explicitly, several times.) In the “Conceptual
Explanations” I begin each section by working plain-old-number-fraction
problems (simplifying them, multiplying them, adding them, and so on); tell
them they can look there if they want a quick review.
Because of the nature of this unit—no new concepts, and fraction phobia—
it has fewer “creative thinking” types of problems, and more “drill and
practice,” than any other unit. It gets boring for you, but don’t let them see
that. For a few students at least, this has the potential to break down a
barrier that they have been struggling with since the third grade.
Rational Expressions -- Rational Expressions
Begin by explaining what rational expressions are, and making the points |
made above—we are going to put together some of our old skills in a new
way, which will require us to be good at fractions. So, we’re going to start
by reviewing how to work with fractions.
Then pair the students up. We’re going to do a sort of do-it-yourself TAPPS
exercise. One partner is the student, one is the teacher. The teacher’s job is
to add 5 + +—not just to come up with the answer, but to walk through
the process, explaining what he is doing and why he is doing it at every
step, all on paper. The student asks for clarifications of any unclear points.
By the time they are done, they should have a written, step-by-step
instruction guide for adding fractions.
Then they switch roles. The former student becomes the new teacher, and
gives two lessons: how to multiply (3) (+) and how to divide y / +. Once
again, they should wind up with a step-by-step guide.
The original teacher takes over again, and shows how to simplify a fraction.
Finally, you give a brief lesson on multiplying fractions—they’ ve already
done that, but the key point to emphasize here is that you can cancel before
you multiply. For instance, if you want to multiply £ times aia you could
Say:
10 _ _70_
ot * 168
and then try to simplify that. But it’s a lot easier to simplify before you
multiply. The a becomes + the 2 becomes 2, so we have:
5
A a
Bt, 12
4
Of course, you can only do this trick—canceling across different fractions
—when you are multiplying. Never when you are adding, subtracting, or
dividing!
So why are we going through all this? Because, even though they know
how to do it with numbers, they are going to get confused when it comes to
doing the exact same thing with variables. So whenever they ask a question
(“What do I do next?” or “Do I need a common denominator here?” or
some such), you refer them back to their own notes on how to handle
fractions. I have had a lot of students come into the test and immediately
write on the top of it:
They did this so they would have a “template” to follow when adding
rational expressions—it’s a very smart move.
Other than basic fraction manipulation, there is only one other big thing to
know about rational expressions—always factor first. Factoring shows you
what you can cancel (especially when multiplying), and how to find the
least common denominator (when adding or subtracting).
So, walk through some sample problems for them on the blackboard. Put up
the problem, and ask them what the first step is...and what the second step
is...and so on, until you have something like this on the blackboard.
sl NG oe Se A DRT Ce Se
ee ray a =
Emphasize over and over that this is just the same steps you would take to
add 5 + 2 But at the end, point out one other thing—just as we did in the
very first week of class, we have asserted that two functions are equal. That
means they should come out exactly the same for any x and y. So have
everyone choose an x-value and a y-value, and plug them into both J =f ;
y+2x
xy
and , and make sure they come out with the same number.
Now walk through something harder on the blackboard, like this:
x _ _ 2x
22+5x+6 23—9x
x i 2x
(x+3)(x+2) a(x+3)(z—3)
eh hs __
(a+3)(2+2) (2+3)(x—3)
z(z—3 2(r+2)
(x? —3x)—(2x+4)
(2+2)(a+3) (2-3)
e?—5x—4
(a+2)(x+3)(x—3)
The original problem
Always factor first!
Simplify (cancel the “ax”
terms on the right).
Get a common denominator.
This step requires a lot of
talking through. You have to
explain where the common
denominator came from, and
how you can always find a
common denominator once
you have factored.
Now that we have a common
denominator, we can
combine. This step is a very
common place to make errors
—by forgetting to
parenthesize the (2% + 4) on
the right, students wind up
adding the 4 instead of
subtracting it.
Done! Of course, we could
multiply the bottom through,
and many students want to. I
don’t mind, but I don’t
recommend it—there are
advantages to leaving it
factored.
Once again, have them try numbers (on their calculators) to confirm that
x 2x : e?—5x—4
2245x416 reper gives the same answer as (@-+2)(@-+3)(x—3)
they should also remember (from week 1) how to find the domain of a
function—and in this case, the two are not quite the same. The function we
ended up with excludes x = —2, x = —3, and x = 3. The original function
excludes all of these, but also x = O. So in that one case, the two are not
identical. For all other cases, they should be.
for any x. But
Whew! OK, you’ve been lecturing all day. If there are 10 minutes left, they
can begin the exercise. They should work individually (not in groups or
pairs), but they can ask each other for help.
Homework:
Finish the in-class exercise and do “Homework—Rational Expressions”
Rational Expressions -- Rational Equations
After you have answered all the questions on the previous homework, they
can just get started on this assignment immediately— it should explain itself
pretty well.
However, after about 5 minutes—when everyone has gotten past the first
two problems—pull them back and talk to the whole class for a moment,
just to make sure they get the point. The point is that if the denominators
are the same, then the numerators must be the same (#1); and if the
denominators are not the same, then you make them the same (#2). It’s
pretty straightforward with these two problems, but it may be deceptively
easy. The real thing to make sure they “get” is that, having established these
two principals with these easy problems, they are now going to apply them
in much more complicated ones.
Then they can get back to it, and you just float around and help. In #3, the
only trick is remembering that if 2? = 25, then x = +5 (not just 5).
Numbers 4 is straightforward. Give them time to struggle with #5 before
pointing out that they should factor-and-simplify first—always factor first!
#6 is really what all this is building up to: to solve rational equations in
general, you must be able to solve quadratic equations!
Homework:
“Homework: Rational Expressions and Equations”
Polynomial Division
A teacher's guide to polynomial division.
Half the class, maybe the whole class, will be lecture today—you have to
show them how to do this. The lecture goes something like this.
Today we’re going to talk about everybody’s favorite topic...long division!
Before we do that, I have to start by pointing out some very important cases
where you don’t have to use long division. For instance, suppose you have
this:
Equation:
362° + 8x? + 5x +10
2x
That problem doesn’t require any hard work—you should be able to divide
it on sight. (Have them do this.) You should have gotten:
18x? + 4x + 25 4 2
x
To take another example, how about this?
Equation:
a> — 6x" + 5x
xz? — 5x
That one isn’t quite as easy. What do we do first? (factor!) Oh yes, let’s do
-. 2«(x—5)(x—1)
| Slee AC asi
that! So it is aa —5)
division necessary.
. Oh, look...it’s just ze — 1! Once again, no long
OK, but suppose we had this?
Equation:
6x? — 8x? 4+ 4x — 2
2x —4
How can we simplify it? This is where we’re going to have to use...long
division.
So, let’s start the same way we started with fractions: by remembering how
to do this with numbers. Everyone, at your seats, work out the following
problem on paper.
Equation:
4327
(Here you pause briefly while they work it on paper—then you work it on
the blackboard.) OK, you should have gotten 393 with a remainder of 4. So
the actual answer is 393 a. How could we check that? That’s right...we
would want to make sure that 393 re times 11 gives us back 4327. Because
that’s what multiplication is—it’s division, backward.
Now, let’s go back to that original problem. OK, kids...I’m going to leave
my number long division over here on the blackboard, and I’m going to
work this rational expressions long division next to it on the blackboard, so
you can see that all the steps are the same.
We’ |l start here:
2x-4| 6xL8x2+4x-2
From there, develop the whole thing on the blackboard, step by step. (I do
this exact problem in the “Conceptual Explanations.” I would not suggest
you tell them to look it up at this point; instead, I would recommend that
you go through it on the blackboard, explaining steps as you go, and
continually reinforcing the analogy to what you did with numbers.) In the
3 9,2 pe
end you conclude that 8*—8* +42
2x—4
22: or, to put it another way, 322 +22+6+
is 322 + 2x + 6 with a remainder of
22
2x+4 °
So then you ask: OK, how could we test that? Hopefully they will come up
with one answer, then you say “Good, how else?” and they come up with
the other one. One way is to plug a number—any number—into
6x? —8x?+4x—2 22
2x—4 2x—4?
sure you get the same thing. The other way is to multiply back
327 + 27+ 655 by 2a — 4 and make sure you get back to
, and the same number into 327 +22 +6 and make
6a° — 8x7 + 4x — 2. Make sure they understand both ways. If you don’t
understand the first way, you don’t understand function equality; if you
don’t understand the second way, you don’t understand what division is!
Finally, you ask: how could we have made all that a bit easier? The answer,
of course, is that we should have divided the top and bottom by 2 before we
did anything else. This brings us back to our cardinal rule of rational
expressions: always factor first!
If there is still time left in class, let them get started on the homework.
Homework:
“Dividing Polynomials”
When going over it, see how many people did #1 the “hard way.” Remind
them that if the bottom is only one term, you can just do the whole thing
quickly and painlessly!
Optional Exercise:
If you have extra time—if some students get way ahead and you want to
give them an extra assignment, or if you want to spend more time on this
topic—here is a good exercise that brings things together. Suppose you
want to solve the equation 6x* — 5x? — 41x — 30 = 0. You can use the
“Solver” on the calculator and it will find one answer: most likely, = —1.
(This requires a 5-minute introduction to the “Solver.”) How do you find
the other answers?
Well, we recall from our study of quadratic equations that if ¢ = —1 is an
answer, then the original function must be expressible as
(x + 1)(something). How do you find the something? With long division!
(Maybe have them try it both ways.) What you end up with, after you
divide, is 62? — 11x — 30. You can factor that the “old-fashioned” way
(which takes a bit of time) and you get (2x2 + 3)(3a — 10) which gives you
the other two roots.
Time for Another Test!
And we’re done with yet another unit.
On #6 of the sample test, stress that they will get no credit without showing
work. They can check their answer either way—multiplying back, or trying
a number—but they have to show their work.
The extra credit is a good problem that I like to get in somewhere. I
generally give one point for the obvious pairs (0,0) and (2,2), and two
points for the equation zy = x + y. But what I really want to see is if they
x
remember how to solve that for y and get y = —;. Finally, from that, they
should be able to see that z = 1 has no pairing number (which is obvious if
you think about it: nothing plus one gives you the same thing times one!).
Introduction
An introduction to the teacher's guide on radicals.
Well, this is easier, isn’t it? They know what a radical is. But they’re going
to go places with them that they have definitely never been before...
This looks long, but a lot of it is very fast. You don’t have to do much setup,
except to remind them what a square root is. I would explain it by analogy
to the way we explained logs. log,8 asks the question “2 to what power is
8?” Well, \/9 also asks a question: “What squared is 9?”
But then, there is an important distinction—one that I like to make right
away, and then repeat several times. The question “what squared is 9?”
actually has two answers. So if we defined square root as the answer to that
question, square root would not be a function—9 would go in, and both 3
and —3 would come out (the old “rule of consistency” from day 1). So we
somewhat arbitrarily designate the ie symbol to mean the positive answer,
so that it is a function. So if you see x? = 9 you should properly answer
x = +3. But if you see x = V9 then you should answer only x = 3. This
is a subtle distinction, but I really want them to get it, and to see that it is
nothing inherent in the math—just a definition of the square root, designed
to make it single-valued. This is why if you see 2” = 2 you have to answer
z= +vV2,to get both answers.
So, on to the assignment. It starts with a couple of word problems, just to
set up the idea that radicals really are useful (which is not obvious). After
everyone is done with that part, you may want to ask them to make up their
own problems that require square roots as answers (they are not allowed to
repeat #1). Get them to realize that we square things all the time, and that’s
why we need square roots all the time, whenever we want to get back.
(We’ll be returning to this theme a lot.)
Then there are problems with simplifying radicals. For many of them, they
have seen this before—they know how to turn v8 into 2 V2. But they
don’t realize that they are allowed to do that because of the general rule that
Vab= Jav b. So it’s important for them to get that generalization, but it’s
also important for them to see that it allows them to simplify radicals. And
it’s equally important for them to see that Va + b is not \/at+ Vb.
The final question is a trap, of course—many of them will answer x*. But
they should know they can test their answer by squaring back, and oops,
eA )isnobe:”,
There are two ways to look at this problem correctly. One is that Vv x16 asks
a question: “What number, squared, is x'°? The rules of exponents are
enough to answer this with x®. The other way to look at it is to remember
that raising something to the + power is the same as taking a square root.
So V 216 is the same as (a'®) z which, again by the rules of exponents, is
a. (I prefer the first way.)
Homework:
“Homework: Radicals”
When going over the homework the next day, make sure to talk about the
last few problems (the inverse functions). Make sure they tested them!
There are several points that you want to make sure they got.
e x” has no perfect inverse. ,/x works only if we confine ourselves to
positive numbers. On the other hand, ¥/z is a perfect inverse of x3. I
always take a moment here to talk about what ¥/ z is, and also to make
sure they understand that it is not the same thing as 3 4/2, so it’s
vitally important to be careful in how you write it.
e «x° and 3z are completely different functions. This is why exponents
really have two different inverses, logs and radicals. Our last unit was
on one of them, this unit will be on the other.
Radicals -- A Bunch of Other Stuff About Radicals
Yeah, it’s sort of a grab bag—a miscellaneous compilation of word
problems, review from yesterday, and so on. You can just get them started
working on the assignment after you’re done going over yesterday’s
homework.
Toward the end, however, they are going to start running into trouble. This
is when you introduce rationalizing the denominator. You may want to
bring the whole class together to see who can figure out how to rationalize
aa . It’s a great opportunity to review our rules of multiplying binomials:
(a = b) 2 — q? + 2ab + b? which is why multiplying by / 3 + 1 doesn’t
work; (a — b)? = a? — b? which is why multiplying by V3 — 1 does.
But please be very careful here, because this particular topic has a very
subtle danger. A lot of teachers communicate the idea that denominators
should always be rationalized, “just because”—because I said so, or
because somehow ‘is “simpler” than re This is one of the best ways to
convince students that math just doesn’t make sense.
What I’m trying to do with this exercise is demonstrate a real practical
benefit of rationalizing the denominator, which is that it helps you add and
subtract fractions. It’s difficult or impossible to come up with a common
denominator without doing this first!
And of course, we have the “you never understand a function until you’ve
graphed it” question. Talk a bit about the graph after they get it. They
should be able to see that the domain and range are both > 0 and why this
must be so. They should see that for large values, it grows very slowly (like
a log), but without the drastic behavior that the log shows in the 0 < z < 1
range.
Homework:
“Homework: A Bunch of Other Stuff About Radicals”
Radical Equations
A teacher's guide to radical equations.
If you read over the assignment carefully, I think it’s pretty self-explanatory.
Encourage the students to read it carefully as they go (not just skip to the
equations).
In my experience, most the trouble with this section comes from trying to
make easy problems, hard (that is, squaring both sides when you don’t need
to); or from trying to make hard problems, easy (neglecting to square both
sides when you should). Make sure they are clear on the distinction—if
there is a variable under the radical, you will need to square; otherwise, you
won't.
Also, it can’t hurt to say this about a hundred times: whenever you square
both sides, you have to check your answers—they may not work even if
you did all your math right! Make sure they understand when this rule
applies, and also why squaring both sides can introduce false answers. (I
work through that explanation pretty carefully in the “Conceptual
Explanations.”)
Homework:
“Homework: Radical Equations”
Time for another test!
Not much to say here, except that the real point of the extra credit is to see
if they realize that the behavior will be very similar on the right, but it will
extend down to the left as well.
Introduction
Some basics on imaginary numbers.
This is an interesting unit in several ways, both good and bad.
The good news is, it’s fun. It’s like a game, and I always try to present it
that way.
The bad news is, it’s incredibly abstract. It’s abstract because it’s hard to
understand these numbers-that-aren’t-numbers, and it’s also abstract
because, to save my life, I can’t come up with any good explanation of why
imaginary numbers are useful. Of course, they are useful—invaluable even
—but how can I explain that to an Algebra II student? Here are a few things
I do always say (several times).
i
Z
These numbers are indeed useful, and they are used in the real world,
even if I can’t do a great job of explaining why to you right now.
Nothing in the real world is imaginary. That is, you will never have i
tomatoes, or measure a brick that is 52 feet long, or wait for 32 + 2
seconds. So why are these useful? Because there are very often
problems where the problem is real, and the answer is real, but in
between, as you get from the problem to the answer, you have to use
imaginary numbers. Repeat this several times. You have a problem, or
real-world situation, which (of course) involves all real numbers. You
do a bunch of math, which includes imaginary numbers. In the end,
you wind up with the answer, which (of course) involves all real
numbers again. But it would have been difficult or impossible to find
that answer, if you didn’t have imaginary numbers.
. One example is electrical engineering. In an electric circuit you have
resistors, capacitors, and inductors. They all act very differently in the
circuit. When you model the circuit mathematically, to determine how
it will behave (what current will flow through it), the inductor has an
inductance and the capacitor has a capacitance and the resistor has a
resistance and they are all very different, which makes the math really
hairy. However, you can define a complex quantity called impedance
which makes resistors, capacitors, and inductors all look
mathematically the same. The disadvantage is that you are now
working with a complex number instead of all real numbers. The
advantage is that resistors, capacitors, and inductors now look the
same in the equations, which makes life a whole lot simpler. So this is
a good example of how you use complex numbers to make the math
easier. (As a side note, electrical engineers call the imaginary number 7
whereas everyone else on the planet calls it 2. I think kids like that bit
of trivia.)
4. Imaginary numbers are also used in many other applications, such as
quantum mechanics.
It’s all very hand-wavy, and I admit that up front, and I don’t hold the kids
responsible for it. But I want them to know that this is really useful, and at
the same time, I want to explain why we aren’t going to have any “real
world” problems in this unit.
This lecture, by the way, usually comes toward the end of day 1, or during
day 2—not at the very beginning of day 1. In the beginning, I prefer to treat
it as a game—“What if there were a square root of —1? Just suppose, what if
there were?” With one class I went so far in treating it as a game that it was
day 3 before they realized I wasn’t making the whole thing up.
Oh yeah, one more thing. The calculators will do imaginary numbers for
them. I never tell them this. If they figure it out, more power to them. But I
literally don’t tell them until after the test that the calculator knows anything
at all about imaginary numbers! I want them to be able to do these things on
their own.
Introduction to Imaginary Numbers
An introduction to the teacher's guide on imaginary numbers.
This is a fun day, or possibly two days. The first exercise is something that,
in theory, they could walk all the way through on their own. But it sets up
all the major themes in imaginary numbers.
In practice, of course, some groups will have problems, and will need help
at various points. But beyond that, almost no groups will see the point of
what they have done, even if they get it right. So a number of times in class,
you are going to interrupt them and pull them back together into a classwide
discussion, and discuss what they have just done. The ideal time to do this
is after everyone in the class has reached a certain point—for instance, after
they have all done #2 (or struggled with it in vain), you pull them back and
talk about #2. All my suggested interruptions are described below.
Before you start, remind them that the equation x? =— 1 has no answer, and
talk about why. Then explain that we are going to pretend it has an answer.
The answer is, of course, an “imaginary” number, so we will call it 2. The
definitions of z is therefore i = /—1 or, equivalently, i? =—1.
There are two ways to play this. One is to go into the whole “why i is
useful” spiel that I spelled out above. The other approach, which is the one I
take, is to treat it as a science fiction exercise. I always start by telling the
class that in good science fiction, you start with some premise: “What if
time travel were possible?” or “What if there were a man who could fly?”
or something like that. Then you have to follow that premise rigorously,
exploring all the ramifications of that one false assumption. So that is what
we are going to do with our imaginary number. We are going to start with
one false premise: “What if you could square something and get —1?” And
we are going to follow that premise logically, using all the rules of math,
and see where it would lead us.
Then they get started. And in #2, they get stopped in their tracks. So you
give them a minute to struggle, and then walk it through on the board like
this.
¢ —z means —1 e2 (*Stress that this is not anything unusual about 2, it is
a characteristic of —1. We could just as easily say —2 means —1 e 2, and
so on. So we are treating 2 just like any other number.)
e Soi(—i) isie—1 ei.
e But we can rearrange that as 7 e 2 e —1. (You can always rearrange
multiplication any way you want.)
¢ Butz ez is —1, by definition. So we have — 1e—1, so the final answer is
1.
The reason to walk through this is to get across the idea of what I meant
about a science fiction exercise. Everything we just did was simply
following the rules of math—except the last step, where we multiplied 2 e 2
and got —1. So it illustrates the basic way we are going to work: assume that
all the rules of math work just like they always did, and that 2? = —1.
The next few are similar. Many of them will successfully get / —25on their
own. But you will have to point out what it means. So, after you are
confident that they have all gotten past that problem (or gotten stuck on it),
call the class back and talk a bit. Point out that we started out by just
defining a square root of -1. But in doing so, we have actually found a way
to take the square root of any negative number! There are two ways to see
this answer. One is (since we just came off our unit on radicals) to write
J —25 = V25- —1 = V—25V/—1 = 5i. The other—which I prefer—is to
say, \/—25 is asking the question “What number squared is -25? The
answer is 52. How do you know? Try it! Square 52 and see what you get!”
Now remind them of the subtle definition of square root as the positive
answer. If you see the problem x? = —25 you should answer x = +5i
(take a moment to make sure they all got the right answer to #4, so they see
why (—5)? gives -25). On the other hand, /—25 is just 5i.
Next we move on to the cycle of powers. Again, they should be able to do
this largely on their own. If a group needs a hint, remind them that if we
made a similar table with powers of 2 (2, 27, 2°, and so on), we would get
from each term to the next one by multiplying by 2. So they should be able
to figure out that in this case, you get from each term to the next by
multiplying by 2, and they should be able to do the multiplication. They will
see for themselves that there is a cycle of fours. So then you can ask the
whole class what 74° must be, and then 74! and so on, and get them to see
the general algorithm of looking for the nearest power of 4. (I have tried
mentioning that we are actually doing modulo 4 arithmetic and I have
stopped doing this—it just confuses things. I do, however, generally
mention that the powers of —1 go in a cycle of 2, alternating between 1 and
—1, so this is just kind of like that.)
#13 is my favorite “gotcha” just to see who falls into the trap and says it’s
9-16.
After they do #18, remind them that this is very analogous to the way we
got square roots out of the denominator. And this is not a coincidence—~ is
a square root, after all, that we are getting out of the denominator! You may
want to introduce the term “complex conjugate” even at this stage, but the
real discussion of complex numbers will come later.
Homework:
“Homework: Imaginary Numbers”
When going over this homework the next day, make sure they got the point.
Our “pattern of fours” can be walked backward as well as forward. It
correctly predicts that i? = 1 which it should anyway, of course, since
anything? = 1. It correctly predicts that i~! = —i which is less obvious—
but remind them that, just yesterday, they showed in class that - simplifies
to =1!
Complex Numbers
A teacher's guide to complex numbers.
The first thing you need to do is define a complex number. A complex
number is a combination of real and imaginary numbers. It is written in the
form a + bi, where a and b are both real numbers. Hence, there is a “real
part” (a) and an “imaginary part” (bi). For instance, in 3 + 43, the real part
is 3 and the imaginary part is 47.
At this point, I like to try to put this in context, by talking about all the
different kinds of numbers we have seen. We started with counting
numbers: 1, 2, 3, 4, and so on. If you are counting pebbles, these are the
only numbers you will ever need.
Then you add zero, and negative numbers. Are negative numbers real
things? Can they be the answers to real questions? Well, sure...depending
on the question. If the question is “How many pebbles do you have?” or
“How many feet long is this stick?” then the answer can never be —2:
negative numbers are just not valid in these situations. But if the answer is
“What is the temperature outside?” or “How much money is this company
worth?” then the answer can be negative. This may seem like an obvious
point, but I’m building up to something, so make sure it’s clear—we have
invented new numbers for certain situations, which are completely
meaningless in other situations. I also stress that we have gone from the
counting numbers to a more general set, the integers, which includes the
counting numbers plus other stuff.
Then we add fractions, and the same thing applies. If the question is “How
many pebbles do you have?” or “How many live cows are on this farm?”
the answer can never be a fraction. But if the question is “How many feet
long is this stick?” a fraction may be the answer. So again, we have a new
set—the rational numbers—and our old set (integers) is a subset of it. And
once again, these new numbers are meaningful for some real life questions
and not for others. I always mention that “rational numbers” (“rational” not
meaning “sane,” but meaning rather a “ratio”) are always expressable as the
ratio of two integers, such as + or a So since we have already defined
the integers, we can use them to help define our larger set, the rational
numbers.
But some numbers are not rational—they cannot be expressed as the ratio
of two integers. These are the irrational numbers. Examples are 7, e, and
/2 (or the square root of any other number that is not a perfect square).
If we add those to our collection—put the rational and irrationals together—
we now have all the real numbers. You can draw a number line, going
infinitely off in both directions, and that is a visual representation of the real
numbers.
And now, finally, we have expanded our set even further, to the complex
numbers, a + 62. Just as we piggybacked the definition of rational numbers
on top of our definition of integers, we are piggybacking our definition of
complex numbers on top of our definition of real numbers.
All that may sound unnecessary, and of course, it is. But some students
really get into it. I have had students draw the whole thing into a big Venn
diagram—which I did not ask them to do. (*It is a good extra credit
assignment, though.) My own diagram is at the very end of this unit in the
“Conceptual Explanations,” under the heading “The World of Numbers.”
Many students like seeing all of math put into one big structure. And it
helps make the point that complex numbers—just like each other
generalization—are valid answers to some questions, but not to others. In
other words, as I said before, you will never measure a brick that is 5i
inches long. (It never hurts to keep saying this.)
The complex numbers are completely general—any number in the world
can be expressed as a + bi. This is not obvious! There are plenty of things
you can write that don’t look like a + bz. One example is + which does not
look like a + 62 but can be put into that form, as we have already seen.
Other examples are 2’ and In(z), which we are not going to mess with, but
they are worth pointing out as other examples of numbers that don’t look
like a + 62, but take-my-word-for-it you can make them if you want to.
And then there is \/ z, which we are going to tackle tomorrow.
That is probably all the setup you need. They can do the in-class exercise on
Complex Numbers and see for themselves that whether you add, subtract,
multiply, or divide them, you get back to a complex number. Also make
sure they get the point about what it means for two complex numbers to be
equal: this will be very important as we move on. One other thing I like to
mention at some point (doesn’t have to be now) is that there are no
inequalities with imaginary numbers. You cannot meaningfully say that
1 > 72 or that 1 < 7. Because they cannot be graphed on a number line, they
don’t really have “sizes”—they can be equal or not, but they cannot be
greater than or less than each other.
Homework:
“Homework: Complex Numbers”
When going over this homework, make a special point of talking about #12.
This helps reinforce the most important point of the year, about
generalizations. Once you have found what happens to (a + bi) when you
multiply it by its complex conjugate, you have a general formula which can
be used to multiply any complex number by its complex conjugate, without
actually going through the work. Show them how #6, 8, and 10 can all be
solved using this formula. Also, remind them that a and 6 are by definition
real—so the answer a? + b? is also real. That is, whenever you multiply a
number by its complex conjugate, you get a real answer. This is why there
is no possible answer to #14.
Me, Myself, and the Square Root of i
A teacher's guide to the square root of the imaginary number.
This is arguably the most advanced, difficult thing we do all year. But I like
it because it contains absolutely nothing they haven’t already done. It’s not
here because it’s terribly important to know V/i, or even because it’s terribly
important to know that all numbers can be written in a + 62 format. It is
here because it reinforces certain skills—squaring out a binomial (always a
good thing to practice), working with variables and numbers together,
setting two complex numbers equal by setting the real part on the left equal
to the real part on the right and ditto for the imaginary parts, and solving
simultaneous equations.
Explain the problem we’re going to solve, hand it out, and let them go.
Hopefully, by the end of class, they have all reached the point where they
know that aa + --iand- 1
1 .
— —— ——7 are the two answers, and have tested
V2 V2 V2 v2
them.
Note that right after this in the workbook comes a more advanced version of
the same thing, where they find 7 ek (all three answers: —1, > -- v3 5 and
_ 2
oe 3 z). I tried using this for the whole class, and it was just a bridge too
far. But you could give it to some very advanced students—either as an
alternative to the i exercise, or aS an extra credit follow-up to it.
Homework:
They should finish the worksheet if they haven’t done so, including #7.
Then they should also do the “Homework: Quadratic Equations and
Complex Numbers.” It’s a good opportunity to review quadratic equations,
and to bring in something new! (It’s also a pretty short homework.)
When going over the homework, make sure they did #3 by completing the
Square—again, it’s just a good review, and they can see how the complex
answers emerge either way you do it. #4 is back to the discriminant, of
course: if b? — 4ac < 0 then you will have two complex roots. The answer
to #6 is no. The only way to have only one root is if that root is 0. (OK, 0 is
technically complex...but that’s obviously not what the question meant,
right?)
The fun is seeing if anyone got #5. The answer, of course, is that the two
roots are complex conjugates of each other—real part the same, imaginary
part different sign. This is obvious if you rewrite the quadratic formula like
this:
_ =b /b2—4ac
mar +t 2a
and realize that the part on the left is always real, and the part on the right is
where you get your z from.
Time for another test!
Not much to say here, except that you may want to reuse this extra credit on
your own test—if they learn it from the sample (by asking you) and then get
it right on your test, they learned something valuable.
Matrices -- Introduction
This is a “double” unit—that is, it is so long that I have a major test right in
the middle of it.
The EOC spends an inordinate amount of time on problems like this:
The Kind of Problem I Don’t Bother Too Much With
This matrix shows McDonald’s sales for a three-day period.
Big Macs Fries Coke
Monday $1,000 $500 $2,000
Tuesday $1,500 $700 $2,700
Wednesday $800 $800 $1,500
What were their total sales on Monday? What were their total sales of Big
Macs? On which day did they make the most profit? etc etc...
I guess the object is to make it appear that “matrices are useful” but it is
really deceptive. Of course, matrices are useful, but not because they give
you a convenient way to organize tabular data and then add columns or look
things up.
So I don’t spend much time on this kind of thing. I start with what a matrix
is (which is sort of like that). I develop the rules for adding matrices,
subtracting them, multiplying a matrix by a constant, and setting two
matrices equal to each other—all of which are very obvious, and should not
be presented as a mystery, but rather just as something obvious.
Then comes the big two days of magic, in which we learn to multiply
matrices. I use a “gradebook” application which gives an example of why
you would want to do this strange operation—it makes a lot of sense up to
the point where you are multiplying an arbitrary-dimensions matrix by a
column matrix, although it gets a bit strained when you expand the second
matrix. No matter. They need to get the mechanics of how you multiply
matrices, and just practice them.
After that bit of magic, the rest should follow logically. The definition of [J]
, the definition of an inverse matrix and how you find one, and (the final
hoorah) the way you use matrices to solve linear equations, should all be
logical and consistent, based on the one magic trick, which is multiplying
them. Oh, also there is a magic trick where you find determinants, which
doesn’t have much to do with anything else.
There is one other thing I need to address, which is calculators. There is a
day that I set aside to teach them explicitly how to do matrices on the
calculator. But that day is after the first test. Before that day, I don’t
mention it at all. And even after that day, I stress doing things by hand, and
give them problems that will force them to do so (by using variables). But I
do love showing them that you can solve five equations with five unknowns
quickly and easily by using matrices and a calculator!
Introduction to Matrices
Tell them to get into groups and work on “Introduction to Matrices.” I think
it is very self-explanatory. You may want to make the analogy at some point
that setting two matrices equal to each other is kind of like setting two
complex numbers equal to each other: for “this” to equal “that,” all their
respective parts must be equal.
Homework:
“Homework—Introduction to Matrices”
Matrices -- Multiplying Matrices I (row x column)
Once again, you may just want to get them started on the assignment, and
let it speak for itself.
However, toward the end of class—after they have all struggled through, or
gotten stuck—pull them back and do some blackboard talk. Start by
pointing out that we are doing two different things here. Multiplying a
matrix times a constant is both easy and intuitive. If you can add two
matrices, then you can add [A] + [A] and thus figure out what 2[A] has to
be.
On the other hand, multiplying two matrices (a row times a column) is not
easy or intuitive. Show how to multiply a row matrix by a column matrix,
do a few examples, and talk about how it applies to the gradebook example.
In other words, make sure they get it.
What I do, about a hundred times, is try to get them to visualize the row
floating up in the air and twisting around so that it lines up with the column.
I use my hands, I use sticks, anything to get them to visually see this row
floating up and twisting to line up with that column. This visualization is
not essential today, but if they get it today, it will really help them
tomorrow, when we start multiplying full matrices! As I mentioned earlier
—this is a radical departure from my normal philosophy—I am more
concerned that they get the mechanics here (how to do the multiplication)
than any sort of logic behind it. They should be able to see, for instance,
that if the row and column do not have the same number of elements, then
the matrix multiplication is illegal.
Oh yeah, one more thing—I always stress that when you multiply two
matrices, the product is a matrix. In the case of a row times a column, it is a
1 x 1 matrix, but it is still a matrix, not a number.
Homework
“Homework—Multiplying Matrices I”
Multiplying Matrices II (the full monty)
The second part in a teacher's guide to multiplying matrices.
This time, you’re going to have to lecture. You are going to have to explain,
on the board, how to multiply matrices. Probably a good 20 minutes (half
the class) dedicated to showing them that this row goes over here to this
column, and then we go down to the next row, and so on. Get them to work
problems at their desks, make sure they are cool with it. You can also refer
them to the “Conceptual Explanations” to see a problem worked out in a
whole lot of detail.
Two things to stress:
1. Keep doing the visualization of a row (in the first matrix) floating up
and twisting to get next to a column (in the second matrix). If the two
do not line up—that is, they have different numbers of elements—then
the multiplication is illegal.
2. Matrix multiplication does not commute. If you switch the order, you
may turn a legal multiplication into an illegal one. Or, you may still
have a legal multiplication, but with a different answer. AB and BA
are completely different things with matrices.
You may never get to the in-class assignment at all. If you don’t, that’s OK,
just skip it! However, note that the in-class assignment is built on one
particular application, which is showing how Professor Snape can do just
one matrix multiplication to get the final grades for all his students. This
exercise is one of the few applications I have for matrix multiplication.
Homework:
“Homework—Multiplying Matrices II”
#4 is important for a couple of reasons. First, of course, by using variables,
it forces them to do the work manually even if they have figured out how to
do it on a calculator. More importantly, it continues to hammer home that
message about what variables are—you can solve this leaving x, y, and z
generic, and then you can plug in numbers for them if you want.
#5 and #7 set up the identity matrix; #6 sets up using matrices to solve
linear equations. You don’t need to mention any of that now, but you may
want to refer back to them later. I don’t want them to think of [I] as being
defined as “a diagonal row of 1s.” I want them to know that it is defined by
the property AI = AI = A, and to see how that definition leads to the
diagonal row of 1s. #7 is the key to that.
Matrices -- Identity and Inverse Matrices
This may, in fact, be two days masquerading as one—it depends on the
class. They can work through the sheet on their own, but as you are
circulating and helping, make sure they are really reading it, and getting the
point! As I said earlier, they need to know that [I] is defined by the property
AI = IA = A, and to see how that definition leads to the diagonal row of
1s. They need to know that A~! is defined by the property AA~' = A!= J
, and to see how they can find the inverse of a matrix directly from this
definition. That may all be too much for one day.
I also always mention that only a square matrix can have an [J]. The reason
is that the definition requires J to work commutatively: AI and IA both
have to give A. You can play around very quickly to find that a 2 x 3
matrix cannot possibly have an |J] with this requirement. And of course, a
non-square matrix has no inverse, since it has no [J] and the inverse is
defined in terms of [J]!
Homework:
“Homework—The Identity and Inverse Matrices”
Matrices -- Inverse of the Generic 2x2 Matrix
This is one of those things that should be easy, but it isn’t. It should be easy
because they have already been doing it, with numbers, and it’s just the
same with letters. But hey, that’s what Algebra II is about, right?
They should definitely work in groups here. Make sure they understand
what they are doing. A clear sign that they don’t understand what they are
doing, even a little, is that they wind up solving for a, or solving for w in
terms of x, or something like that. They need to understand that the object is
to solve for w, x, y, and z in terms of a, b, c, and d. Only by doing this can
they come up with a generic solution to the inverse of a 2x2 matrix, which
can then be used quickly and easily to find the inverse of any 2x2 matrix. If
they don’t understand that, they just don’t have any idea what we’re doing
—it’s important to get them to understand the problem instead of just
focusing on solving it.
ad—be —¢
this form, help them understand how to use it—the numbers in this diagonal
switch places, the number in that diagonal change signs. This also helps set
up the determinant (ad-bc), by the way.
ad —b
The answer, by the way, is A-1= = | | . Once they have it, in
a
Matrices -- Use Matrices for Transformation
This is a fun day. No new math, just a cool application of the math we’ve
seen.
The in-class assignment pretty well speaks for itself. It’s worth mentioning
that, despite the very simplified and silly nature of this specific assignment,
the underlying message—that matrices are used to transform images in
computer graphics—is absolutely true.
0
Some students (good students) may question why the last column iH is
necessary in specifying Harpoona’s initial condition. The reason is this
representation is not simply a list of all the comer points in a shape. Yes,
each column represents a point. But the matrix is a set of instructions to the
computer, to draw lines from this point to that point. Without the last
column, Harpoona would be missing her hypotenuse.
Homework
Homework: “Homework: Using Matrices for Transformation.” Here we see
a matrix that rotates any object by 300, counter-clockwise. The inverse
matrix, of course, rotates clockwise. Hopefully they will discover all that
for themselves. And hopefully most of them will realize why an inverse
matrix always does the exact opposite of the original matrix, since if you do
them one after the other, you end up back where you started.
Time for Another Test!
Our first test on matrices.
Matrices -- Matrices on the Calculator
This starts with a lecture. You have to show them how to do matrices on the
calculator. They should be able to...
e Enter several matrices at once
e Add several matrices
e Subtract
e Multiply
e Find inverses
All of this is explained step-by-step in the “Conceptual Explanations.”
There are two things I stress. First, whenever I enter a matrix, I always
check it. For instance, after I enter matrix [A], I go back to the home screen
and go [A][Enter], and the calculator displays matrix [A] to me, so I can
make sure I typed it right. One small mistype will ruin a whole problem,
and it’s really easy to do!
Second, the calculator is very smart about interpreting equations. After you
enter three matrices, you can just type [A][B]-1-[C] and it will multiply [A]
by the inverse of [B] and then subtract [C].
Homework
“Homework—Calculators.” Depending on how things go, they may be able
to finish this in class and have no homework.
Matrices -- Determinants
Another very lecture-heavy topic, I’m afraid. Like multiplying matrices,
finding the determinant is something you just have to show on the board.
And once again, you can refer them in the end to the “Conceptual
Explanations” to see an example worked out in detail.
Start by talking about the ad-bc that played such a prominent role in the
inverse of a 2X2 matrix. This is, in fact, the determinant of a 2x2 matrix.
Then show them how to find the determinant of a three-by-three matrix,
using either the “diagonals” or “expansion by minors” method, whichever
you prefer. (I would not do both. Personally, I prefer “expansion by
minors,” and that is the one I demonstrate in the “Conceptual
Explanations.”)
Hot points to mention:
e Brackets like this [A] mean a matrix; brackets like this |A] mean a
determinant. A determinant is a number associated with a matrix: it is
not, itself, a matrix.
e Only square matrices have a determinant.
e Also show them how to find a determinant on the calculator. They
need to be able to do this (like everything else) both manually and with
a calculator.
e To find the area of a triangle whose vertices are (a,b), (c,d), and (e,f),
a © -e
you can use the formula: Area=%%| 6b d_f |. This is the only use I
ae ee
can really give them for determinants. They will need to know this for
the homework. Do an example or two. I like to challenge them to find
that area any other way, just to make the point that it is not a trivial
problem without matrices. (I don’t know any other good way.)
e All we’re really going to use “expansion by minors” for is 3x3
matrices. However, I like to point out that it can be obviously extended
to 4x4, 5x5, etc. It also extends down to a 2x2—if you “expand
minors” on that, you end up with the good old familiar formula ad-bc.
e Finally, mention that any matrix with determinant zero has no inverse.
This is analogous to the rule that the number 0 is the only number with
no inverse.
Homework
“Homework—Determinants”
Matrices -- Solving Linear Equations
Just let them start this assignment, it should explain itself.
This is the coolest thing in our whole unit on matrices. It is also the most
dangerous.
The cool thing is, you can solve real-world problems very quickly, thanks to
matrices—so matrices (and matrix multiplication and the inverse matrix and
so on) prove their worth. If you are given:
26 — oy teal
64 —-y+2=4
4x — 10y+22=2
2 -—-5 1 ii
you plug into your calculator [A] = 6 -—1 2 ,[B])= 4 ,ask for
4 -—-10 2 p
A~'B, and the answer pops out! They should be able to do this process
quickly and mechanically.
But the quick, mechanical nature of the process is also its great danger. I
want them to see the logic of it. I want them to see exactly why the equation
2:5 1 & 1
6 -1l 2 y = 4 is exactly like those three separate equations up
4 -10 2 z 2
there. I want them to be able to solve like this:
AX=B
Aree AEB
IX = A!B
X=A"'B
to see why it comes out as it does. If they see that this is all perfectly
logical, and they know how to do it, the day is a big success—and in fact,
this sort of justifies the whole unit on matrices.
As a final point, mention what happens if the equations were unsolvable:
matrix A will have a 0 determinant, and will therefore have no inverse, so
the equation won’t work. (You get an error on the calculator.)
Homework
“Homework—Solving Linear Equations”
Time for Another Test!
And, time to conclude our unit on matrices.
Introduction
Felder Algebra 2 Teacher's Guide introduction to chapter on modeling data
with functions.
This unit is really three different topics, joined together by a somewhat
weak thread.
The first topic is direct and inverse variation. The second topic is finding a
parabola that fits any three given points. The third topic is regression on a
calculator.
The weak thread that connects them is that they all involve starting with
data, and finding a function that models that data. (At least, the latter topics
are Clearly about that, and the first topic is about that if you approach it the
way I do...)
Modeling Data with Functions -- Direct and Inverse Variation
This is one of those days where you want to get them working right away
(on the “Direct Variation” assignment), let them finish the assignment, and
then do 10-15 minutes of talking afterwards. You want to make sure they
got the point of what they did.
Direct variation is, of course, just another kind of function—an independent
variable, and a dependent variable, and consistency, and so on. But the
aspect of direct variation that I always stress is that when the independent
number doubles, the dependent number doubles. If one triples, the other
triples. If one is cut in half, the other is cut in half...and so on. They are, in
a word, proportional.
This is not the same thing as saying “When one goes up, the other goes up.”
Of course that is true whenever you have direct variation. But that statement
is also true of In(x), —_, x2, 2x, x+3, and a lot of other functions: they do
“when one goes up, the other goes up” but not “when one doubles, the other
doubles” so they are not direct variation. The only function that has that
property is f(x)=kx, where k is any constant. (Point out that k could be 4, or
it could be —2, or any other constant—not just a positive integer.)
In #3 they arrived at this point. They should see that the equation y=kx has
the property we want, because if you replace x with 2x then y=k(2x) which
is the same thing as 2kx which is twice what it used to be. So if x doubles, y
doubles.
They should also see that direct variation always graphs as a line. And not
just any line, but a line through the origin.
But the thing I most want them to see is that there are many, many
situations where things vary in this way. In other words, #4 is the most
important problem on the assignment. You may want to ask them to tell the
whole class what they came up with, and then you throw in a few more, just
to make the point of how common this is. The amount of time you spend
waiting in line varies directly with the number of people in front of you; the
amount you pay varies directly with the number of meals you order; the
weight of your french fries measured in grams varies directly with the
weight of your french fries measured in pounds; and both of these, in turn,
vary directly with the number of fries; and so on, and so on, and so on.
Homework
Part I
“Homework: Inverse Variation”
That’s right, no homework on direct variation—it’s time to develop the
second one. They should be able to do it pretty well on their own, in
analogy to what happened in class. But you will spend a fair amount of the
next day debriefing them on inverse variation, just as you did on direct. The
defining property is that when the independent variable doubles, the
dependent variable chops in half. Again, it is true to say “when one goes up,
the other goes down”—but it is not enough. 1/x2 has that property, and so
does 10-x, and neither of those is inverse variation.
Examples are a bit harder to think of, off the top of your head. But there is
an easy and systematic way to find them. I always warn the students that I
will ask for an example of inverse variation on the test, and then (now that I
have their attention) I explain to them how to do it. Inverse variation is
y=k/x (graphs as a hyperbola). This equation can be rewritten as xy=k. This
is useful for two reasons. First, it gives you the ability to spot inverse
variation—if the product is always roughly the same, it’s inverse. Second, it
gives you the ability to generate inverse variation problems, by thinking of
any time that two things multiply to give a third thing, and then holding that
third thing constant.
For instance: the number of test questions I have to grade is the number of
students, times the number of questions on the test. That’s obvious, right?
If I want to turn that into a direct variation problem, I hold one of the two
multiplying variables constant. What do I mean “hold it constant?” I mean,
pick a number. For instance, suppose there are twenty students in my class.
Now the dependent variable (number of questions I have to grade) varies
directly with the independent variable (number of questions I put on the
test).
On the other hand, if I want to turn that exact same scenario into an inverse
variation problem, I hold the big variable constant. For instance, suppose I
know that I am only capable of grading 200 problems in a night. So I have
to decide how many questions to put on the test, based on how many
students I have. You see? Double the number of students, and the number of
questions on the test drops in half.
Algebraically, if I call t the number of questions on the test, s the number of
students, and g the number of questions I have to grade, then g=ts. In the
first case, I set s=20 so I had the direct variation equation g=20t. In the
second case I set g=200 so I had the inverse variation equation t=200/s.
I explain all that to my class, slowly and carefully. They need to know it
because the actual test I give them will have a question where they have to
make up an inverse variation problem, and I always tell them so. That
question, if nothing else, will come as no surprise at all.
Part II
“Homework: Direct and Inverse Variation”
This is on the long side, and introduces a few new ideas: the idea of being
proportional to the square (or square root) of a variable, and the idea of
dependence on multiple variables. So you should ideally hand it out in the
middle of class, so they have time to work on it before the homework—and
be prepared to spend a lot of time going over it the next day.
Modeling Data with Functions -- Calculator Regression
Lecture time. Talk about the importance of regression again. Walk them,
step-by-step, through a few regressions on the calculator. (Detailed
instructions for a TI-83 are included in the “Conceptual Explanations” for
this section.)
My advice right now is to look at the points first (which may require
resetting the window!) and then categorize them according to concavity—
although I don’t use that word. I talk about three kinds of increasing
functions:
e Linear functions increase steadily
e Logs and square roots increase more and more slowly
e Parabolas and exponential functions increase more and more quickly
Similarly, of course, for decreasing functions. Choose an appropriate kind
of regression, and let the calculator do the rest.
After a few examples, hand out the homework.
Homework
“Homework: Calculator Regression”
Time for Another Test!
Question 11c is not something I would put on a real test, for a couple of
reasons. First, it would be pretty hard to grade; second, and more
importantly, it is unlike anything we’ve seen on a homework. But I might
use it for an extra credit, and in any case, it can’t hurt them to see it and
discuss it on a sample.
Conic Sections Guide -- Introduction
The last topic! And it’s a big one. Here is the overall plan.
We start with a day on distance—finding the distance between two points,
finding the distance from a point to a line—this will form a basis for the
whole unit.
Then we do our shapes: circles, parabolas, ellipses, and hyperbolas. For
each shape, there are really two things the students need to learn. One is the
geometric definition of the shape. The other is what I call the machinery
associated with the shape—the standard equation and what it represents. I
cover these two things separately, and then connect them at the end.
Distance
A teacher's guide to lecturing on distance in preparation for later classes on
conic sections.
Before I hand this out, I tell them a bit about where we’re going, in terms of
the whole unit. We’re going to do “analytic geometry”—that is, linking
geometry with algebra. We’re going to graph a bunch of shapes. And
everything we’re going to do is built upon one simple idea: the idea of
distance.
Let’s start with distance on a number line. (Draw a number line on the
board.) Here’s 4, and here’s 10. What’s the distance between them? Right,
6. You don’t need to do any math, you can just count—1, 2, 3, 4, 5, 6. There
we are.
How about the distance from 4 to 1? Right, 3. 4 to 0? Right, 4. One more:
what is the distance from 4 to -5? Again, just count...1,2,3,4,5,6,7,8,9. The
distance is nine.
So, what’s going on here? In each case, we’re counting from 4 to some
other number: to 10, to 1, to 0, then to -5. What happened mathematically?
We subtracted. This is a point that looks incredibly obvious, but really it
isn’t, so it’s worth repeating: if you subtract two numbers, you get the
distance between them. 10-4 is 6. 4-1 is 3. And the last one?
4 — (—5) = 4+ 5 = 9. So even that works. Remember that we saw, by
counting, that the distance from 4 to -5 is 9. Now we see that it works
mathematically because subtracting a negative is like adding a positive.
Oh yeah...what if we had subtracted the other way? You know, 4-9 or -5-4.
We would have gotten the right answers, only negative. But the distance
would still be positive, because distance is always positive.
So, based on all that, at your seats, write down a formula for the distance
from a to b on a number line: go! (give them thirty seconds) What did you
get? | — |? Good! | — |? Also good! They are the same thing. - and
- are not the same thing, but when you take the absolute value, then they
are.
Now, let’s get two-dimensional here. We’ lI start with the easy case, which is
when the points line up. In that case, we can use the same rule, right? For
instance, let’s look at (4,3) and (10,3). How far apart are they? Same as
before—6. We can just count, or we can just subtract, because the y-
coordinates are the same. (Show them this visually!) Similarly, suppose we
take (-2,5) and (-2,-8). Since the -coordinates are the same, we can just
count again, or just subtract the -coordinates, and get a distance of 13.
Now, what if neither coordinate is the same? Then it’s a bit trickier. But
we’re not going to use any magic “distance formula”—if you ever
memorized one, throw it out. All we need is what we’ve already done. Let’s
look at (-2,1) and (4,9). (Draw it!) To find that distance, we’re going to find
the distance across and the distance up. So draw in this other point at (4,1).
Now, draw a triangle, with the distance we want over here, and the distance
across here, and the distance up here. These two sides are easy, because
they are just what we have already been doing, right? So this is 6 and this is
8. So how do we find this third side, which is the distance we wanted?
Right, the Pythagorean Theorem! So it comes out as 10.
The moral of the story is—-whenever you need to find a distance, use the
Pythagorean Theorem.
OK, one more thing before you start the assignment. That was the distance
between two points.How about the distance from a point to a line? For
instance, what is the distance from you to the nearest street? The answer, of
course, is—it depends on where on the street you want to get. But when we
say, distance from you to the street, we mean the shortest distance. (Do a
few drawings to make sure they get the idea of shortest distance from a
point to a line. If the line is vertical or horizontal, then we are back to
just counting. If it’s diagonal, life gets much more complicated, and
we’re not going to get into it. Except I sometimes assign, as an extra
credit assignment, “find the distance from the arbitrary point ( , ) to
the arbitrary line = + .It’s ugly and difficult, but I usually have
one or two kids take me up on it. For the rest of the class, just promise
to stick with horizontal and vertical lines, and counting.)
After all that is said, they are ready to start on the assignment.
Homework:
“Homework: Distance”
Circles
A teacher's guide to lecturing on circles.
Our first shape. (Sometimes I have started with parabolas first, but I think
this is simpler.)
Here’s how you’re going to start—and this will be the same for every
shape. Don’t tell them what the shape is. Instead, tell them this. A bunch
of points are getting together to form a very exclusive club. The
membership requirement for the club is this: you must be exactly 5 units
away from the origin. Any point that fulfills this requirement is in the
club; any point that is either too close or too far, is not in the club. Give
them a piece of graph paper, and have them draw all the points in the club.
You come around and look at their work. If they are stuck, point out that
there are a few very obvious points on the x-axis. The point is that, before
any math happens at all, every individual group should have convinced
themselves that this club forms a circle.
Then you step back and say—in Geometry class you used circles all the
time, but you may never have formally defined what a circle is. We now
have a formal definition: a circle is all the points in a plane that are the
same distance from a given point. That distance is called what? (Someone
will come up with “radius.”) And that given point is called what? (Someone
may or may not come up with “center.”) The center plays a very interesting
role in this story. It is the most important part, the only point that is key to
the definition of the circle—but it is not itself part of the circle, not itself a
member of the club. (The origin is not 5 units away from the origin.) This is
worth stressing even though it’s obvious, because it will help set up less
obvious ideas later (such as the focus of a parabola).
Point out, also, that—as predicted—the definition of a circle is based
entirely on the idea of distance. So in order to take our general geometric
definition and turn it into math, we will need to mathematically understand
distance. Which we do.
At this point, they should be ready for the in-class assignment. They may
need some help here, but with just a little nudging, they should be able to
see how we can take the geometric definition of a circle leads very directly
to the equation of acircle, x A y k r where (h,k) is the
center and 7 is the radius. This formula should be in their notes, etc. Once
we have this formula, we can use it to immediately graph things like
x y —to go from the equation to the graph, and vice-
versa.
Once you have explained this and they get it, they are ready for the
homework. They now know how to graph a circle in standard form, but
what if the circle doesn’t come in standard form? The answer is to
complete the square—twice, once for x and once for y. If it works out that
the homework gets done in class (this day or the next day), you may want to
do a brief TAPPS exercise with my little completing-the-square demo in the
middle of the homework. However, they should also be able to follow it on
their own at home.
Homework:
“Homework: Circles”
It may take a fair amount of debriefing afterward, and even a few more
practice problems, before you are confident that they “get” the circle thing.
They need to be able to take an equation for a circle in non-standard form,
put it in standard form, and then graph it. And they need to still see how
that form comes directly from the Pythagorean Theorem and the definition
of a circle.
There is one other fact that I always slip into the conversation somewhere,
which is: how can you look at an equation, such as
ab bb y y , and even tell if it is a circle? Answer: it has
both an xz anday term, and they have the same coefficient. If there is no
x ory term, you have a line. If one exists but not the other, you have a
parabola. By the time we’re done with this unit, we will have filled out all
the other possible cases, and I expect them to be able to look at any
equation and recognize immediately what shape it will be.
Parabolas, Day 1
A teacher's guide to lectures on parabolas.
Once again, when you start, don’t tell them we’re doing parabolas! Tell
them we’re going to create another club. This time the requirement for
membership is: you must be exactly the same distance from the point (0,3)
that you are from the line . For instance, the point (3,3) is not part
of our club—it is 3 units away from (0,3) and six units away from
Now, let them work in groups on “All the Points Equidistant from a Point
and a Line” to see if they can find the shape from just that. If they need a
hint, tell them there is one extremely obvious point, and two somewhat
obvious points. After that they have to dink around.
When all or most groups have it, go through it on the blackboard,
something like this. The extremely obvious point is the origin. The
“somewhat” obvious points are (-6,3) and (6,3). Show why all those work.
Now, can any point below the x-axis work? Clearly not. Any point below
the x-axis is “obviously” (meaning, after you show them for a minute)
closer to the line, than to the point.
So, let’s start working up from the origin. The origin was in the club. As we
move up, we are getting closer to the point, and farther away from the
line. So how can we maintain equality? The only way is to move farther
away from the point, by moving out. In this way, you sketch in the
parabola.
Now, you introduce the terminology. We’re already old friends with the
vertex of a parabola. This point up here is called the focus. This line down
here is the directrix. The focus and directrix are kind of like the center of a
circle, in the sense that they are central to the definition of what a parabola
is, but they are not themselves part of the parabola. The vertex, on the other
hand, is a part of the parabola, but is not a part of the definition.
The directrix, of course, is a horizontal line: but what if it isn’t? What is the
directrix is vertical? Then we have a horizontal parabola. Of course it isn’t
a function, but it’s still a shape we can graph and talk about, and we have
seen them a few times before. If you have time, work through
Homework:
“Homework: Vertical and Horizontal Parabolas”
Parabolas, Day 2
A teacher's guide to lectures on parabolas.
What good are parabolas? We’ve already seen some use for graphing
parabolas, in terms of modeling certain kinds of behavior. If I throw a ball
into the air, not straight up, its path through the air is a parabola. But here is
another cool thing: if parallel lines come into a parabola, they all bounce to
the focus.
Draw this
on the
board:
So telescopes are made by creating parabolic mirrors—all the incoming
light is concentrated at the focus. Pretty cool, huh?
We are already somewhat familiar with parabola machinery. We recall that
the equation for a vertical parabola is y = a(x — h)? + k and the equation
for a horizontal parabola is 2 = a(y — k)? + h. The vertex in either case is
at (h,k). We also recall that if a is positive, it opens up (vertical) or to the
right (horizontal); if a is negative, it opens down or to the left. This is all old
news.
Thus far, whenever we have graphed parabolas, we have found the vertex,
determined whether they open up or down or right or left, and then gone
“swoosh.” If we were a little more sophisticated, we remembered that the
width of the parabola was determined by a; so y = 2x? is narrower than
y = x” which is narrower than y = sa’. But how can we use a to actually
draw the width accurately? That is the question for today. And the answer
starts out with one fact which may seem quite unrelated: the distance from
the vertex to the focus is x: I’m going to present this as a magical fact for
the moment—we will never prove it, though we will demonstrate it for a
few examples. But that one little fact is the only thing I am going to ask you
o “take my word for”—everything else is going to flow logically from that.
(Go back to your drawing of our parabola with focus (0,3) and directrix
y = —8, and point out the distance from vertex to focus—label it 4S )
Now, what is the eines from the vertex to the directrix? Someone should
get this: it must also be 7~. Why? Because the vertex is part of the
parabola, so by definition. it must be the same distance from the directrix
that it is from the focus. Label that.
Now, let’s start at this point here (point to (-6,3)) and go nevi to ie
directrix. al long is that? Again, someone should get it: z— + —— , which
is — or =~. Now, how far over is it, from this same point - the focus?
Well, sain. this is on the parabola, so it must be the same distance to the
focus that it is to the directrix: oe
If oo enend that line all the way to the other side, you have two of those:
sx a erie 2, or — = , running between (6,3) and (—6,3). This line is called
the jaine er meee for “straight line”—it is always a line that touches
the parabola at two points, runs parallel to the directrix, and goes through
the focus.
At this point, the blackboard looks something like this:
2a
pola
oil
da ‘i.
: fe 2a
da
The point should be clear. Starting with the one magical fact that the
distance from the focus to the vertex is ~ we can get everything else,
including the length of the latus rectum, +. And that gives us a way of
doing exactly what we wanted, which is using a to more accurately draw a
parabola.
At this point, you do a sample problem on the board, soup to nuts. Say,
C= + (y oa 3) 2 _ 4. They should be able to see quickly that it is
horizontal, opens to the right, with vertex at (-4,-3)— that’s all review. But
1
now we can also say that the distance from the focus to the vertex is z—
which in this case (since a = +) is +: So where is the focus? Draw the
parabola quickly on the board. They should be able to see that the focus is
to the right of the vertex, so that puts us at(—34,—3). The directrix is to
the left, at x = 4s. (I always warn them at this point, a very common
way to miss points on the test is to say that the directrix is (2 = —4+,—3).
That’s a point—the directrix is a line!)
How wide is the parabola? When we graphed them before, we had no way
of determining that: but now we have the latus rectum to help us out. The
length of the latus rectum is ~ which is 2. So you can draw that in around
the focus—going up one and down one—and then draw the parabola more
accurately, staring from the focus and touching the latus rectum on both
sides.
Whew! Got all that? Good then, you’re ready for the homework! (You may
want to mention that a different sample problem is worked in the
“Conceptual Explanations.)
Homework:
“Homework: Parabolas and the Latus Rectum”
This one will take a lot of debriefing afterwards. Let’s talk about #3 for a
minute. The first thing they should have done is draw it—that can’t be
stressed enough—always draw it! Drawing the vertex and focus, they
should be able to see that the parabola opens to the right. Since we know
the vertex, we can write immediately x = a(y — 6)? + 5. Many of them
will have gotten that far, and then gotten stuck. Show them that the distance
from the vertex to the focus is 2, and we know that this distance must be —
, SO — — 2. We can solve this to find a = ~ which fills in the final piece
of the puzzle.
On #5, again, before doing anything else, they should draw it! They should
see that there is one vertical and one horizontal parabola that fits this
definition.
But how can they find them? You do the vertical, they can do the
horizontal. Knowing the vertex, we know that the vertical one will look like
y = a(x + 2)? + 5. How can we find a this time, by using the information
that it “contains the point (0,1)? Well, we have to go back to our last unit,
when we found the equation for a parabola containing certain points!
Remember, we said that if it contains a point, then that point must make the
equation true! So we can plug in (0,1) to the equation, and solve for a.
1=a(0+2)?+5, 4a +5 =1, a = —1. The negative a tells us that the
parabola will open down—and our drawing already told us that, so that’s a
good reality check.
Parabolas: From Definition to Equation
A teacher's guide on teaching the connection between the definition and
equation of a parabola, and how to get from one to the other.
OK, where are we? We started with the geometric definition of a parabola.
Then we jumped straight to the machinery, and we never attempted to
connect the two. But that is what we’re going to do now.
Remember what we did with circles? We started with our geometric
definition. We picked an arbitrary point on the circle, called it (x,y), and
wrote an equation that said “you, Mr. (zx, y), are exactly 5 units away from
the origin.” That equation became the equation for the circle.
Now we’re going to do the same thing with a parabola. We’re going to
write an equation that says “you, Mr. (a, y), are the same distance from the
focus that you are from the directrix.” In doing so, we will write the
equation for a parabola, based on the geometric definition. And we will
discover, along the way, that the distance from the focus to the vertex really
ae
1S Ae *
That’s really all the setup this assignment needs. But they will need a lot of
help doing it. Let them go at it, in groups, and walk around and give hints
when necessary. The answers we are looking for are:
iL / x2 + y?. As always, hint at this by pushing them toward the
Pythagorean triangle.
2.y + 4. The way I always hint at this is by saying “Try numbers.
Suppose instead of (a, y) this were (3,10). Now, how about (10,3)?”
and so on, until they see that they are just adding 4 to the y-coordinate.
Then remind them of the rule, from day one of this unit—to find
distances, subtract. In this case, subtract -4.
3, \/x2 + y? = y+ 4. This is the key step! By asserting that d1 = d2,
we are writing the definition equation for the parabola.
4. Good algebra exercise! Square both sides, the y? terms cancel, and
you’re left with 2 = 8y + 16. Solve for y, and you end up with
C= oe — 2. Some of them will have difficulty seeing that this is the
final form—point out that we can rewrite it as y = i (a — 0) oat
that helps. So the vertex is (0,-2) and the distance from focus to vertex
is 2, just as they should be.
Warn them that this will be on the test!
Which brings me to...
Time for another test!
Our first test on conics. I go back and forth as to whether I should give them
a bunch of free information at the top of the test—but it’s probably a good
idea to give them a chart, sort of like the one on top of my sample.
Ellipses
A teacher's guide to lecturing on ellipses.
Only two shapes left! But these two are doozies. Expect to spend at least a
couple of days on each—they get a major test all to themselves.
In terms of teaching order, both shapes are going to follow the same pattern that
we set with parabolas. First, the geometry. Then, the machinery. And finally, at
the end, the connection between the two.
So, as always, don’t start by telling them the shape. Let them do the assignment
“Distance to this point plus distance to that point is constant” in groups, and
help them out until they get the shape themselves. A good hint is that there are
two pretty easy points to find on the z-axis, and two harder points to find on the
y-axis. As always, keep wandering and hinting until most groups have drawn
something like an ellipse. Then you lecture.
The lecture starts by pointing out what we have. We have two points, called the
foci. (One “focus,” two “foci.”) They are the defining points of the ellipse, but
they are not part of the ellipse. And we also have a distance, which is part of the
definition.
Because the foci were horizontally across from each other, we have a horizontal
ellipse. If they were vertically lined up, we would have a vertical ellipse. You
can also do diagonal ellipses, but we’re not going to do that here.
Let’s talk more about the geometry. One way you can draw a circle is to
thumbtack a piece of string to a piece of cardboard, and tie the other end of the
string to a pen. Keeping the string taut, you pull all the way around, and you
end up with a circle. Note how you are using the geometric definition of a
circle, to draw one: the thumbtack is the center, and the piece of string is the
radius.
Now that we have our geometric definition of an ellipse, can anyone think of a
way to draw one of those? (probably not) Here’s what you do. Take a piece of
string, and thumbtack both ends down in a piece of cardboard, so that the string
is not taut. Then, using your pen, pull the string taut.
& &
Now, pull the pen around, keeping the string taut. You see what this does?
While the string is taut, the distance from the pen to the left thumbtack, plus
the distance from the pen to the right thumbtack, is always a constant—
namely, the length of the string. So this gives you an ellipse. I think most
people can picture this if they close their eyes. Sometimes I assign them to do
this at home.
OK, so, what good are ellipses? The best example I have is orbits. The Earth,
for instance, is traveling in an ellipse, with the sun at one of the two foci. The
moon’s orbit around the Earth, or even a satellite’s orbit around the Earth, are
all ellipses.
Another cool ellipse thing, which a lot of people have seen in a museum, is that
if you are in an elliptical room, and one person stands at each focus, you can
hear each other whisper. Just as a parabola collects all incoming parallel lines at
the focus, an ellipse bounces everything from one focus straight to the other
focus.
OK, on to the machinery. Here is the equation for a horizontal ellipse, centered
at the origin.
Equation:
Here is a drawing of a horizontal ellipse.
There are three numbers in this drawing. a is the distance from the center to the
far edge (right or left). If you double this, you get the horizontal length of the
entire ellipse—this length, 2a of course, is called the major axis.
b is the distance from the center to the top or bottom. If you double this, you get
the vertical length of the entire ellipse—this length, 26 of course, is called the
minor axis.
c is the distance from the center to either focus.
a and b appear in our equation. c does not. However, the three numbers bear the
following relationship to each other: a? = b? + c?. Note also that a is always
the biggest of the three!
A few more points. If the center is not at the origin—if it is at, say, (h, k), what
do you think that does to our equation? They should all be able to guess that we
replace x? with (a — h)? and y? with (y — k)?.
Second, a vertical ellipse looks the same, but with a and b reversed:
Equation:
Let’s see how all that looks in an actual problem—walk through this on the
blackboard to demonstrate. Suppose we want to graph this:
x? + 9y*- 4x2 + 54y + 49 = 0
First, how can we recognize it as an ellipse? Because it has both an x? anda y
term, and the coefficient are different. So we complete the square twice, sort of
like we did with circles. But with circles, we always divided by the coefficient
right away—this time, we pull it out from the x and y parts of the equation
separately.
a” + 9y?-4e + 54y+ 49 =0
a?—4a + Oy” + 54y = —49
(x? 4a) + 9(y? + 6y) = —49
(a*-4a + 4) + 9(y? + 6y+ 9) = -49+4+81
(x — 2)? + 9(y+ 3)? = 36
The
original
problem
Group the
x and y
parts
Factor out
the
coefficients.
In this case,
there is no
x
coefficient,
SO we just
have to do
y.In
general, we
must do
both.
Complete
the square,
twice
Finish
completing
that square
Divide by
36. This is
because we
need a 1 on
the right, to
be in our
standard
form!
OK, get all that? Now, what do we have?
First of all, what is the center? That’s easy: (2,-3).
Now, here is a harder question: does it open vertically, or horizontally? That is,
does this look like a + ye = 1, or like se + ve = 1? The way to tell is by
remembering that a is always bigger than b. So in this case, the 36 must be a”
and the 4 must be b?, so it is horizontal. We can see that the major axis (2a) will
be 12 long, stretching from (-4,-3) to (8,-3). (Draw all this as you’re doing it.)
And the minor axis will be 4 long, stretching from (2,-5) to(2,-1).
And where are the foci? Since this is a horizontal ellipse, they are to the left and
right of the center. By how much? By c. What is c? Well, a? = b? + c?. So
c? = 32, andc = V32 = 4 V2, or somewhere around 55. (They should be
able to do that without a calculator: 32 is somewhere between 25 and 36, so
1/32 is around 55.) So the foci are at more or less (—3+,-3) and (7+,-3).
We’re done!
Homework:
“Homework: Ellipses”
There are two things here that may throw them for a loop.
One is the fractions in the denominator, and the necessity of (for instance)
2
turning 25y? into TB . You may have to explain very carefully why we do
that (standard form allows for a number on the bottom but not on the top), and
how we do that.
The other is number 7. Some students will quickly and carelessly assume that
94.5 is a and 91.4 is b. So you want to draw this very carefully on the board
when going over the homework. Show them where the 94.5 and 91.4 are, and
remind them of where a, b, and c are. Get them to see from the drawing that
94.5 + 91.4 is the major axis, and is therefore 2a. And that a — 91.4 is c, so
we can find c, and finally, we can use a” = b? + c? to find b. This isa really
hard problem, but it’s worth taking a lot of time on, because it really drives
home the importance of visually being able to see an ellipse in your head, and
knowing where a, 0, and c are in that picture.
Oh, yeah...number 6 may confuse some of them too. Remind them again to
draw it first, and that they can plug in (0,0) and get a true equation.
OK, now we’re at a bit of a fork in the road. The next step is to connect the
geometry of the ellipse, with the machinery. This is a really great problem,
because it brings together a lot of ideas, including some of the work we did
forever ago in radical equations. It is also really hard. So you can decide what
to do based on how much time you have left, and how well you think they are
following you. You may want to have them go through the exercise in class
(expect to take a day). Or, you may want to make photocopies of the
completely-worked-out version (which I have thoughtfully included here in the
teacher’s guide), and have them look it over as a TAPPS exercise, or just ask
them to look it over. Or you could skip this entirely, or make it an extra credit.
Conic Sections Guide -- Ellipses: From Definition to Equation
Here is the geometric definition of an ellipse. There are two points called the
“foci”: in this case, (-3,0) and (3,0) . A point is on the ellipse if the sum of its
distances to both foci is a certain constant: in this case, I’ll use 10 . Note that
the foci define the ellipse, but are not part of it.
The point (x ,y) represents any point on the ellipse. d1 is its distance from the first focus, and d2
to the second. So the ellipse is defined geometrically by the relationship: d1 + d2 = 10.
To calculate d1 and d2, we use the Pythagorean Theorem as always: drop a straight line down
from (x, y) to create the right triangles. Please verify this result for yourself! You should find
that dl = \/(a + 3)? + y? and d2 = \/(x — 3)? + y?. So the equation becomes:
J/(x +3)? + y? + /( — 3)? + y? = 10. This defines our ellipse
The goal now is to simplify it. We did problems like this earlier in the year (radical equations,
the “harder” variety that have two radicals). The way you do it is by isolating the square root,
and then squaring both sides. In this case, there are two square roots, so we will need to go
through that process twice.
————— Isolate a
—S (Gaya
VJ (a +3)? +y? = 10-/(a — 3)? +y radical
2 2 — ies Vase 242 Square
(x +3)? +9? = 100-20,/(e—3)? +9? + (2-3)? +y Se,
————— Multiply
(x? + 62 +9) + y? = 100-20,/(a — 3)? + y? 4 (2? 6x 4 9) ty? out the
squares
Cancel &
12x = 100-20,/(x — 3)? + y? combine
like terms
J/(z — 3)? +y2=5-32 Rearrange,
divide by
20
Square
(x-3)? + y? = 25-62 + 42” both sides
again
Multiply
(x?- 62 + 9) + y? = 25-62 + 3ra” out the
square
16 2 2 __ Combine
ae ty" = 16 like terms
cr Divide by
3 tig =1 16
...and we’re done! Now, according to the “machinery” of ellipses, what should that equation
look like? Horizontal or vertical? Where should the center be? What are a, b, and c? Does all
that match the picture we started with?
Hyperbolas
A teacher's guide to lectures on hyperbolas.
The good news about hyperbolas is, they are a lot like ellipses—a lot of
what has already been learned, will come in handy here. The bad news
about hyperbolas is, they are a lot like ellipses—so all the little differences
can be very confusing.
We will start as always with the geometric definition. Let them do the
assignment “Distance to this point minus distance to that point is constant”
in groups, and help them out until they get the shape themselves. There are
two pretty easy points to find on the x-axis, but from there they just sort of
have to noodle around like we did with parabolas, asking...what happens as
I move inside? What happens as I move outside? As always, keep
wandering and hinting until most groups have drawn something like a
hyperbola. Then you lecture.
The lecture starts by pointing out what we have. We have two points, once
again called the foci. They are the defining points of the hyperbola, but they
are not part of the hyperbola. And we also once again have a distance which
is part of the definition.
Because the foci were horizontally across from each other, we have a
horizontal hyperbola. If they were vertically lined up, we would have a
vertical hyperbola. You can also do diagonal hyperbolas—anyone
remember where we have seen one of those? That’s right, inverse variation!
That was a hyperbola, just like these. But we’re not going to talk about
those in this unit, just the horizontal and vertical ones.
Incidentally, a hyperbola is not two back-to-back parabolas. It looks like it,
but these shapes are actually different from parabolic shapes, as we will see.
OK, so, what good are hyperbolas? The analogy continues...orbits!
Suppose a comet is heading toward the sun. (Draw.) If it has a low energy—
that is, a low velocity—it gets trapped by the sun, and wins up orbiting
around the sun in an elliptical orbit. But if it has high energy (high velocity)
it zooms around the sun and then zooms away forever. Its path in this case
is half a hyperbola.
Another cool use is in submarine detection. A submarine sends out a pulse.
Two receiving stations get the pulse. They don’t know what direction it
came from or when it was sent, but they do know that one station received
it exactly two seconds before the other one. This enables them to say that
the distance from the sub to this station, minus the distance to this other
station, is such-and-such. And this, in turn, locates the sub on a hyperbola.
OK, on to the machinery. Here is the equation for a horizontal hyperbola,
centered at the origin.
Equation:
Looks familiar, doesn’t it? But the plus has changed to a minus, and that
makes all the difference in the world.
Here is a drawing of a horizontal hyperbola.
i
+ + +
ee $b
Let’s be very careful in seeing how this is, and is not, like an ellipse.
a is defined in a very similar way. It goes from the center, to the edges. In
this case, the edges are called the “vertices” (in analogy to parabolas). The
distance from one vertex to the other (2a of course) is called the transverse
axis.
c is defined in a very similar way: it goes from the center to either focus.
b is perpendicular to the other two, just as before. But it goes from the
center to... well, to a sort of strange point in the middle of nowhere. We’re
going to use this point. The distance from the top point to the bottom point (
2b) is called the conjugate axis.
But here is one major difference. In an ellipse, the foci are inside; in a
hyperbola, they are outside. So you can see, just looking at an ellipse, that
a > c; and you can see, just looking at a hyperbola, that c > a. Hence, our
equation relating the three shapes is going to be different. Instead of
a” = b? + c?, we have c? = a? + b?. This reflects the fact that c is the
biggest one in this case.
Once again, the class should be able to see that if the center is (h, k) instead
of the origin, we replace x? with (a — h)? and y” with (y — k)?.
How about a vertical hyperbola? That looks like this:
Equation:
The way we tell vertical from horizontal is completely different. In an
ellipse, we told by assuming that a > b. In a hyperbola, we have no such
guarantee; either a or b could be the greater, or they could even be the same.
Instead, we look at which one comes first. If we are doing an x? — y?
thing, it’s horizontal; if we are doing a y” — x” thing, it’s vertical. This is a
very common source of errors...be careful!
There should be no need to go through the whole completing-the-square
rigmarole on the board—just tell them it is exactly like with ellipses,
including making sure you have a 1 on the right, and there is nothing
multiplied by x? or y?. But the really different part is the graphing. So let’s
just pick it up there. Suppose you want to graph:
(w=2)" (y+)? _y
36 4
First of all, what is the center? That’s easy: (2,-3).
Now, here is a harder question: does it open vertically, or horizontally? We
can answer this question without even looking at the numbers on the
bottom! The x? is positive and the y” is negative, so this is horizontal.
As we did with ellipses, we will then find a, b, and c. a = 6 and b = 2. We
will find c with the hyperbola equation c? = a? + b? (different from the
ellipse equation!) and get c = V/40 which is 2\/10 or somewhere just
above 6 (again, because 40 is just above 36).
Now, it’s drawing time. We start at the center. We go out horizontally by 6
to find the vertices, and by a little more than 6 to find the foci. We go out
vertically by 2 to find the endpoints of the conjugate axis, those weird little
points in space. Then what?
Here’s what you do. Draw a rectangle, going through the vertices, and the
endpoints of the conjugate axis. Then, draw diagonal lines through the
comers of that rectangle. Those diagonal lines are going to serve as
asymptotes, or guides: they are not part of the hyperbola, but they help us
draw it. Why? Because as it moves out, the hyperbola gets closer and closer
to the asymptotes, but never quite reaches them. So once you have drawn
your asymptotes, you have a guide for drawing in your hyperbola.
Note that I draw the box and the asymptotes in dotted lines, indicating that
they are not really part of the hyperbola.
It’s worth talking for a while about what an asymptote is, since it is such an
important concept in Calculus—the line that the curve gets closer and closer
and closer to, without ever quite reaching. It’s also worth pointing out that
this shows that a hyperbola is not two back-to-back parabolas, since
parabolas do not display asymptotic behavior.
Finally, I always mention the comet again. Remember that if a comet comes
in with high energy, it swoops around the sun and then flies away again.
Now, if there were no sun—if there were nothing in the universe but our
comet—the comet would travel in a straight line. And clearly, when the
comet is very, very far away from the sun (either before or after its journey
through our solar system), the effect of the sun is very small, so the comet
travels almost in a straight line. That straight line is the asymptote. The
farther away from the sun the comet gets, the closer it gets to that straight
line.
Homework:
“Homework: Hyperbolas”
The only really unusual thing here is that I ask for the equation of one of the
asymptotes. This is just a quick review of the skill of finding the equation
for a line, given that you already know two points on the line.
You will note that I have absolutely nothing here about going from the
geometry of the hyperbola, to the equation. The reason is that it is exactly
like the ellipse. You may want to do it, or you may not. If you do it, you
shouldn’t need to hand them anything—just say “By analogy to what we
did with the ellipse, do this.”
Now that you have done all the shapes, the one vital skill that cuts across all
of it is looking at an equation (ax? + by? + cx + dy + e = 0) and telling
what shape it is. This is done entirely by looking at the coefficients of the
squared terms (a and 6), and you should refer them to the chart at the end of
the “Conceptual Explanations.”
Time for our very last test!
With luck, you have two or three weeks left for review after this, before the
final test. Congratulations, you made it through!
Sequences and Series Guide -- Prerequisites
The major “prerequisite” for this unit is the introductory unit on functions.
However, the introduction to geometric sequences will work best if the unit
on exponents has already been covered; and the inductive proofs often
require skills covered in the unit on rational expressions.
Sequences and Series Guide -- Arithmetic and Geometric Sequences
The in-class assignment does not need any introduction. Most of them will
get the numbers, but they may need help with the last row, with the letters.
After this assignment, however, there is a fair bit of talking to do. They
have all the concepts; now we have to dump a lot of words on them.
A “sequence” is a list of numbers. In principal, it could be anything: the
phone number 8,6,7,5,3,0,9 is a sequence.
Of course, we will not be focusing on random sequences like that one. Our
sequences will usually be expressed by a formula: for instance, “the xxxnth
terms of this sequence is given by the formula 100 + 3(n — 1)” (or
3n + 97) in the case of the first problem on the worksheet. This is a lot like
expressing the function y = 100 + 3(a — 1), but it is not exactly the same.
In the function y = 3a + 97, the variable x can be literally any number.
But in a sequence, xxxn must be a positive integer; you do not have a
“minus third term” or a “two-and-a-halfth term.”
The first term in the sequence is referred to as ¢; and so on. So in our first
example,¢ = 112.
The number of terms in a sequence, or the particular term you want, is often
designated by the letter n.
Our first sequence adds the same amount every time. This is called an
arithmetic sequence. The amount it goes up by is called the common
difference d (since it is the difference between any two adjacent terms).
Note the relationship to linear functions, and slope.
Exercise:
Problem:
If I want to know all about a given arithmetic sequence, what do I need
to know? Answer: I need to know f, and d.
Exercise:
Problem:
OK, so if I have ¢, and d for the arithmetic sequence, give me a
formula for the n“ term in the sequence. (Answer:
ty, = t; + d(n — 1). Talk through this carefully before proceeding.)
Time for some more words. A recursive definition of a sequence defines
each term in terms of the previous. For an arithmetic sequence, the
recursive definition is t,,1 = t, + d. (For instance, in our example,
tn4i = ty, + 3.) An explicit definition defines each term as an absolute
formula, like the 3n + 97 or the more general t,, = t; + d(n — 1) we came
up with.
Our second sequence multiplies by the same amount every time. This is
called a geometric sequence. The amount it multiplies by is called the
common ratio r (since it is the ratio of any two adjacent terms).
Exercise:
Problem:
Find the recursive definition of a geometric sequence. (Answer:
tn41 = Tt. They will do the explicit definition in the homework.)
Exercise:
Problem:
Question: How do you make an arithmetic sequence go down?
Answer: d < 0
Exercise:
Problem:
Question: How do you make a geometric series go down? Answer:
O0O<r< 1. (Negative r values get weird and interesting in their own
way...why?)
Homework
“Homework: Arithmetic and Geometric Sequences”
Sequences and Series Guide -- Series and Series Notation
Begin by defining a series: it’s like a sequence, but with plusses instead of
commas. So our phone number example of a sequence, “8,6,7,5,3,0,9”
becomes the series “ ” which is 38.
Many of the other words stay the same. The first term is __, the term is
, and so on. If you add up all the terms of an arithmetic sequence, that’s
called an arithmetic series; and similarly for geometric.
The hardest part about this introduction is the notation. Explain about series
notation, using weird examples like — just to make the point that
even when the function looks complicated, it is not hard to write out the
terms.
Note that the “counter” always goes by ones. Does this mean you can’t have
a series that goes up by 2s? Ask them how to use series notation for the
series
Homework:
“Homework—Series and Series Notation”
Sequences and Series Guide -- Arithmetic and Geometric Series
Going over the homework, make sure to mention #3(e), an alternating
series. You get that kind of alternation by throwing in a (—1)” or, in this
case, (=1)""*.
Last night’s homework ended with the series “all the even numbers between
50 and 100.” Some students may have written s37° (48 + 2n). Others
may have written the answer differently. But one thing they probably all
agree on is that adding it up would be a pain. If only there were...a shortcut!
Let’s consider the series 3+5+7+9+4+11+138+4 15+ 17. (Write that
on the board.)
Exercise:
Problem:
What do we get if we add the first term to the last? Answer: 20.
Modify your drawing on the board to look like this:
3+5+7+9+11+13+15417
ee
OK, what about the second term to the second-to-last? Hmm....20 again.
Add to the drawing, and then keep adding until it looks like this:
34+54+7+94+114+13+15+17
=
Exercise:
Problem:
So, looking at that drawing, what does
3+5+7+9+11+13+ 15+ 17 add up to? Hopefully everyone
can see that it adds up to four 20s, or 80.
Exercise:
Problem:
And this is the big one—will that trick work for all series? If so, why?
If not, which series will it work for? Answer: It will work for all
arithmetic series. The reason that the second pair added up the same
as the first pair was that we went up by two on the left, and down by
two on the right. As long as you go up by the same as you go down,
the sum will stay the same—and this is just what happens for
arithmetic series.
OK, what about geometric series? Write the following on the board:
2+6+4+ 18+ 54+ 162 + 486 + 1458
Clearly the “arithmetic series trick” will not work here: 2 + 1458 is not
6 + 486. We need a whole new trick. Here it comes. First, to the left of
your equation, write S' = so the board looks like:
S=2+6+18+4 54+ 162 + 486 + 1458
where Sis the mystery sum we’re looking for. Now, above that, write:
a0.
ask the class what comes next. Can we just multiply each term by 3? (Yes,
distributive property.) When you write this line, line up the numbers like
this:
3S =6+ 184 54+ 162 + 486 + 1458 + 437
S=2+6+18+4 54+ 162 + 486 + 1458
But don’t go too fast on that step—make sure they see why, if S is what we
said, then 3. must be that!
Now, underline the second equation (as I did above), and then subtract the
two equations. What do we get on the left of the equal sign? What do we
get on the right? See how things cancel? See if you can get the class to tell
you that...
2S = 4374-2
So then S is just 2186. They may want to verify this one on their
calculators. Once again, however, the key is to understand why this trick
always works for any Geometric series.
Homework:
“Homework: Arithmetic and Geometric Series”
Sequences and Series Guide -- Proof by Induction
Proof by Induction
Going over last night’s homework, make sure they got the right formulas.
For an arithmetic series, S,, = = (t + ti) For a geometric, S, = ara,
in some form or another.
Example:
Students sometimes ask if that formula will still work for an arithmetic
series with an odd number of terms. Obviously, you can’t still pair them up
in the way we have been doing. The answer is, it does still work. One proof
—which I usually don’t mention unless the right questions are asked—is
that, for an arithmetic series, the average of all the terms is right in the
middle of the first and last terms. (It can take a minute to convince yourself
that this is not always true for any series, but it is for any arithmetic
series.) So the average is (a ) , and there are n terms. This leads us to
the total sum being n(5*), which is the old formula written in a new
way. This lacks some of the elegance of the original proof, but it has the
advantage that it doesn’t matter if n is even or odd.
Anyway, on to today’s topic.
Note:Important note. The following lecture can be done in 5-10 minutes—
I’ve done it many times—and if you do it that way, it doesn’t work. This
can be one of the most confusing topics in the whole unit. It must be taken
very slowly and carefully!
Today, we’re going to learn a new way of proving things. This method,
called “proof by induction,” is a very powerful and general technique that
turns up in many different areas of mathematics: we are going to be
applying it to series, but the real point is to learn the technique itself.
So...to begin with, we are going to prove something we already know to be
true:
14+24+3+4..n= $(n+1)
Of course we know how to prove that using the arithmetic series trick, but
we’re going to prove it a different way.
Let’s start by seeing if that formula works when n = 1: in other words, for
a 1-term series. In that case, what is the left side of the equation? (Even this
seemingly innocuous question can baffle good students sometimes. Give
them a minute. Point to the equation. Remind them that the equal sign
divides any equation into a left side, and a right side. So, what is the left
side of this equation, when there is only one term?) Yes, it is just...1.
How about the right side? Well, that’s... + (1 + 1) = 1. So at least, for this
particular case, it works.
To build up to the next step, ask this hypothetical question. Suppose we had
not yet proven that this equation always works. But suppose that I had
proven that it works when n = 200. Just say, I had sat down with my
calculator and added up all the numbers from 1 to 200, which took a very
long time, but in the end, I did indeed get what the formula predicts (which
is, of course, 100 x 201 = 20, 100). And now I ask you to confirm that the
formula works when n = 201.
Well, you can do the right side easily enough: aye (202) = 20, 301. But
what about the left side? Do you have to add up all those numbers on your
calculator? No, you don’t, if you’re clever. (See if they can figure this next
part out—this is the key.) I already told you what the first 200 numbers add
up to. So you can simply add 201 to my total. 20, 100 + 201 = 20, 301.
The point here is not just “it works.” The point is that you can confirm that
it works, without adding up all 200 numbers again, because I already did
that part—all you have to add is the last number.
Now...suppose I had already proven that it works for n = 325. How would
we show that it works for n = 326? Good—we would add 326 to the old
answer (for n = 325), and see if we got what the formula predicted we
should get for n = 326. Let’s try it...
Now...suppose I had already proven that it works for n = 1000. How
would we show that it works for n = 1001? Good—we would add 1001 to
the old answer (for m = 1000), and see if we got what the formula
predicted we should get for n = 1001. Let’s try it... Repeat this exercise
until they are sick of it, but boy, do they get it. Then hit them with the big
one: what is the general form of this question? See if they can figure out
that it is:
Suppose I had already proven that it works for some n. How would we
show that it works for n + 1?
Give them time here...see if they can find the answer...
We would add (n + 1) to the old answer (for n), and see if we got what the
formula predicted we should get for (n + 1).
What does that look like? Well, for the old n, the formula predicted we
would get + (n ae Ls So if we add (n + 1) to that, we get
7 (n + 1) + (n + 1). And what should we get? Well, for (n + 1), the
formula predicts we should get ae (n + 1+ 1),
Do the algebra to show that they are equal. Then, step back and say...so,
what have we done? Well, first we proved that the formula works for n = 1
. Then we proved—not for one specific case, but quite generally—that if it
works for any number, it must also work for the next number. If it
works for n = 1, then it must work for n = 2. If it works for n = 2, then it
must work for n = 3....and so on. It must always work.
At this point, I think it’s helpful to work through one more example. I
recommend going through 4 ae just as it is done in the Conceptual
Explanations. This time, you’re using a little less explanation and focusing
more on the process, so it makes a better model for their homework.
Homework:
“Homework—Proof by Induction”
At this point, you’re ready for the test. Unlike most of my “Sample Tests,”
this one is probably too short, but it serves to illustrate the sorts of
problems you will want to ask, and to remind the students of what we’ve
covered.
Sequences and Series Guide -- Extra Credit
An extra cool problem you may want to use as an extra credit
or something
Exercise:
Problem:
A bank gives i% interest, compounded annually. (For instance, if
i = 6, that means 6% interest.) You put A dollars in the bank every
year for n years. At the end of that time, how much money do you
have?
Note:(The fine print: Let’s say you make your deposit on January 1
every year, and then you check your account on December 31 of the
last year. So if nm = 1, you put money in exactly once, and it grows for
exactly one year.)
Solution:
The money you put in the very last year receives interest exactly once.
“Receiving interest” in a year always means being multiplied by
(1 + ain): (For instance, if you make 6% interest, your money
multiplies by 1.06.) So the A dollars that you put in the last year is
worth, in the end, A(1 + iis).
The previous year’s money receives interest twice, so it is worth
ae:
A(1 + a) at the end. And so on, back to the first year, which is worth
A (1 a i)" (since that initial contribution has received interest n times).
So we have a Geometric series:
1 4 2 1 n
S=A(1l+7,)+A(QL+ 75) +..4+4A0 4+ =)
We resolve it using the standard trick for such series: multiply the equation
by the common ratio, and then subtract the two equations.
; r 2 ; n j n+1
(eS SA ee ob cde et aA ee)
i i \2 i \n
S=A(1+7,)+A(Ql+ 75) +..4+4A0 4+ =)
(jor) S$ = A(1 + au)
i n+1 i
= 4) (14 ais)" - (1+ ie)
n+1 ;
mAs ets)
Example:
Example: If you invest $5,000 per year at 6% interest for 30 years, you end
up with:
ar") | 1.0631- 1.06] = $419,008.39
Not bad for a total investment of $150,000!
Probability -- Tree Diagrams
You can start them off here with the in-class assignment “How Many
Groups?” or even hand it out after the previous test: it isn’t long or difficult,
and it does not require any introduction.
It does, however, require a lot of follow-up. The worksheet leads to a
lecture, and the lecture goes something like this.
The big lesson for the first day is how to make a chart of all the possibilities
for these kinds of scenarios. For the die-and-coin problem, the tree diagram
looks like (draw this on the board):
die 1 2 3 4 5 6
coin H TT H TH T 4H TH TH T
It may seem silly to repeat “Heads-Tails, Heads-Tails,” and so on, six times.
But if you do so, each “leaf” of this “tree” represents exactly one
possibility. For instance, the third leaf represents “The die rolls a 2, and the
coin gets heads.” That’s a completely different outcome from the fifth leaf,
“The die rolls a 3 and the coin gets heads.”
Using a tree like this, we can answer probability questions.
Exercise:
Problem:
What is the probability of the outcome “Die rolls 2, coin gets heads?”
Just ask this question, give them 15 seconds or so to think about it,
then call on someone for the answer. But then, talk through the
following process for getting the answer.
1. Count the number of leaves that have this particular outcome. In this
case, only one leaf.
2. Count the total number of leaves. In this case, twelve.
3. Divide. The probability is ae
Exercise:
Problem:
What is the probability of the outcome “Die rolls a prime number, coin
gets tails?” Give them 30 seconds or so, then go through it carefully on
the chart. There are three such leaves. (*Trivia fact: 1 is not considered
a prime number.) So the probability is =, or +.
Exercise:
Problem:
What does that really mean? I mean, either you’re going to get that
outcome, or you’re not. After you roll-and-flip, does it really mean
anything to say “The probability of that outcome was +?”
Give the class a minute, in pairs, to come up with the best possible
explanation they can of what that statement, “The probability of this event
is +” really means. Call on a few. Ultimately, you want to get to this: it
doesn’t really mean much, for one particular experiment. But if you repeat
the experiment 1,000 times, you should expect to get this result about 250
of them.
Exercise:
Problem:
Why are there twelve leaves? (Or: How could you figure out that there
are twelve leaves, without counting them?)
Solution:
It’s six (number of possibilities for the die) times two (number of
possibilities for the coin). For the second question on the worksheet,
with the frogs, there are 15,000 groups, or 5,000x3. This multiplication
rule is really the heart of all probability work, so it’s best to get used to
it early.
Get the idea? OK, let’s try a new one.
A Ford dealer has three kinds of sedans: Ford Focus, Taurus, or Fusion. The
Focus sedan comes in three types: S, SE, or SES. The Fusion sedan also
comes in three types: S, SE, or SEL. The Taurus comes in only two models:
SEL, and Limited. (All this is more or less true, as far as I can make out
from their Web site.)
Ask everyone in the class, in pairs, to draw the appropriate tree and use it to
answer the following questions.
e If you choose a car at random from the dealer lot, what are the odds
that it is a Fusion? (Answer: 2)
1
e What are the odds that it is a Fusion SE? (Answer: =.)
e What are the odds that it is any kind of SEL? (Answer: 2)
e Finally—and most important—what assumption must you make in
answering all of the above questions, that was not stated in the original
description of the situation?
Hopefully someone will come up with the answer I’m looking for to that
last question: you’re assuming that the dealer’s lot has exactly the same
number of each possible kind of car. In real life, of course, that assumption
is very likely wrong.
This ties in, of course, to the last question on the assignment they did. Red-
haired people are considerably more rare than the other types. So this
business of “counting leaves” only works when each leaf is exactly as
common, or probable, as each other leaf. This does not mean we cannot do
probability in more complicated situations, but it means we will need to
develop a more sophisticated rule.
Homework
“Homework: Tree Diagrams”
When going over this homework the next day, there are two things you
want to emphasize about problem #2 (stars). First: because the actual tree
diagram would have 70 leaves, you don’t want to physically draw it. You
sort of have to imagine it. They have seen three diagrams now: the coin-
and-die, the cars, and the three-coins. That should be enough for them to
Start to imagine them without always having to draw them.
Second: the answer to question 2(c) is not “one in 70, so maybe 14 or so.”
That logic worked fine with 1(e), but in this case, it is not reasonable to
assume that all types of stars are equally common. The right answer is, “I
can’t answer this question without knowing more about the distribution of
types.”
Problem #3(e) is subtler. The fact that + of the population is children does
not mean that 4 of the white population is children. It is quite possible that
different ethnic groups have different age breakdowns. But ignoring that for
the moment, problem #3 really brings out a lot of the main points that you
want to make the next day:
Probability -- Introduction to Probability
OK, let’s really talk about problem #3 from last night’s homework. And for
the moment, let’s ignore part (e), and go ahead and assume that + of all
white people are children. Based on that assumption, you would expect
roughly 75 white people, about 19 of whom are children, and about 9 of
whom are boys. If you answered exactly e , or 9.375, that isn’t a crazy
answer. Of course, you can’t actually have 9.375 white boys in a room. But
that is actually the “expectation value” for such an experiment. If you have
a thousand rooms with a hundred people each, the average number of white
boys in each room will probably be 9.375.
In the formal language of probability, we would say that for any randomly
chosen person in the U.S. in 2006, there is a 9.375% chance that this person
will be a white boy. That’s what “percent” means: out of a hundred.
Exercise:
Problem: What percent of the people in this class, right now, are girls?
Exercise:
Problem:
If you roll a die, what is the percent chance that you will get an even
number?
OK, that’s easy enough. But we’re going to tweak it a bit. Obviously, there
is nothing magical about the number 100. We could just as easily ask “How
many out of a thousand?” or “How many out of 365?” But what turns out to
be most convenient, mathematically, is to ask the question “How many out
of 1?”
This is how we are going to work with probability numbers from here on
out, so it is very important to understand this numbering system!
e The probability of any event whatsoever, under any and all
circumstances, is always between 0 and 1. A probability of —2, ora
probability of 2, is meaningless.
e A probability of 0 means “It cannot possibly happen.”
e A probability of 1 means “It is guaranteed to happen.”
e A probability of f means “It has a one in four chance of happening,”
or “If you try this 100 times, it will probably happen 25 of them,” so it
is the same as a 25% chance.
After you have said all that, you’re ready to hit them with the worksheet
“Introduction to Probability.” It should only take 10 minutes (half of which
is spent on #2a).
Then come back. Let’s go over #2 carefully.
2b. The probability is +. You can see this from the tree diagram, but how
could we have figured it out without a drawing? The answer—we’ve
discussed this before, and it is absolutely central—is by multiplying. 4
possibilities for the first die, times 4 possibilities for the second die, makes
16 possibilities for the combination.
But here’s another way we can look at that same multiplication. The
probability of “3 on the first die” is +. The probability of “2 on the second
die” is also +. So the probability of both these events happening is + x +
1
or 16°
When you have two different independent events—that is, neither one has
an effect on the other—the probability of both happening is the probability
of the first one, times the probability of the second one.
The idea of “independent” events is crucial here, of course, and you have to
stress it. But it’s also a fairly obvious point, and there is a real danger of
making it sound more esoteric than it is. If you spend ten minutes
discussing the word “independent” you may do more harm than good.
Consider trying this instead. Tell that class that you’re looking into a big
box full of bananas. One out of every four bananas in the box is green; the
rest are yellow. Also, one out of every three bananas is stamped “Ship to
California”; the rest say “ship to New York.” Finally, half the bananas are
over four days old.
e¢ What is the probability that a given banana is green, and destined for
New York? - x + = <- One out of every six bananas have both of
these attributes. Or, to put it another way, a given randomly chosen
banana has a - chance of having both attributes.
e What is the probability that a given banana is green, and over four
days old? Well, not much. Not + x + = +. Because in general, as a
banana gets older, it turns from green to yellow. So being green, and
being old, are not independent: one makes the other less likely.
Now, ask the class, in pairs, to come up with a similar scenario. (It should
not involve fruit!) They should think of two events that are independent,
and calculate the probability of both of them happening. Then they should
think of two events that are not independent, and explain why the
probability of both of them happening is not the product of their individual
probabilities.
Homework
“Homework: The Multiplication Rule”
Going over this homework, of course you want to make sure that the last
problem gets answered. With a little thought, it should be obvious to anyone
that if is the probability that something will occur, 1 — is the
probability that it will not occur. If it happens 1 time out of 5, then it doesn’t
happen 4 times out of 5. This can be memorized as a new rule, along with
the multiplication rule, but it is easier to see why it works.
Trickier Probability Problems
A teacher's guide to difficult probability problems.
The thing that makes probability problems the darling of math contest
writers everywhere, is also the thing that makes them frustrating for so
many students: no two problems are exactly alike. Most probability
problems can be solved with the multiplication rule, combined with a lot of
good, hard thinking about the problem.
I’m going to present two scenarios with five questions here, in the lesson
plan. The idea is for you to talk them through with the class. In each case,
explain the scenario and the question clearly. Then give them a minute or
two, with no guidance, to think about it. Then take their answers and go
over the correct answer very slowly and clearly. None of them should be
presented as if it were a symbol of a whole, unique, important class of
problems. Each should be presented as simply another example of you can
solve a wide variety of problems, if you’re willing to think about them
patiently and clearly.
Example:
Scenario 1
You reach your hand into a bag of Scrabble® tiles. The bag has one tile
with each letter. You pull out, first one tile, and then another.
1. What is the probability that you will pull out, first the letter , and
then the letter ? The quick, easy answer is — —. Quick,
easy....and not quite right. Yes, there is a — chance that the first tile
will bean. But once you have that tile, there are only 25 left. So the
probability of the second tile being a are actually —. The
probability of getting an followedbya are— —.
2. What is the probability that your two tiles are the letters and ? It
looks like the same question, but there is a subtle difference. You
could pull out followed by (as in the last example), or you could
pull out followed by. So there are really two ways to do it, and
the probability is — —
Example:
Scenario 2
You roll two 6-sided dice.
1. What is the probability that the sum of the two dice is 10? Imagine
making a tree diagram. It would have 36 leaves. How many of them
would have a sum of 10? 6—4, 5-5, and 4—6. (Of course, on the tree
diagram, “6 on the first die, 4 on the second” is a different leaf from
“A on the first die, 6 on the second”...just as in the AB problem
above.) So the probability is —, or —.
2. What is the probability that neither die rolls a 1? We do not have a
“neither” rule, so we have to reframe the question in terms of the rules
we do have. We can rephrase the question like this: what is the
probability that the first die doesn’t roll a 1, and the second die also
doesn’t roll a 1? The first is —, and the second is also —. So the
probability of both happening is —. It’s an easy question to answer,
once you reword it correctly.
3. What is the probability that either die (or “at least one die’) rolls a 1?
(Most people think the answer willbe — — _ -—. By that logic, by
the time you roll six dice, you are guaranteed to get at least one 1:
obviously not true!)
The right way to think about this problem is as the reverse, the “not,” of the
previous problem. We said that 25 out of 36 times, neither die will roll a 1.
So the remaining 11 out of 36 times, at least one of them will. This is an
example of the “not” rule we got from last night’s homework: the
probability of “no ones” is —, so the probability of NOT “no ones” is
— —. (*It’s interesting to note that the “naive” guess of — is not
too far off, and makes a reasonable approximation. If you have a 1 in 10
chance of doing something, and you try three times, there is a roughly —
chance that you will succeed at least once—but not exactly —.)
The last thing you need to assure the class, before you hit them with the
worksheet, is that no one is born knowing how to do this. Probability
problems are just like everything else: they make more sense, and get
easier, with practice. It’s OK to get frustrated, but don’t give up!
Then give them the worksheet. Ideally they should be able to make a good
(10-15 minute) start in class, and then finish it up for homework. Expect to
spend a lot of the next day going over these. It’s worth it.
Homework
“Homework: Trickier Probability Problems”
Permutations
A teacher's guide to permutations.
No in-class worksheet today—a day of lecture.
How many different three-digit numbers can we make using only the digits
1, 2, and 3? Answer: 27. Here they are, listed very systematically. (If
possible, project this table onto a screen where everyone can see it and look
at it for a moment, to see the pattern and how it is generated.)
First Second Third Resulting
Digit Digit Digit Number
1 111
1 2 112
3 113
1 121
1 2 2 122
3 123
1 131
3 2 132
3 133
2 1 1 211
First Second Third Resulting
Digit Digit Digit Number
2 212
3 213
ih 221
2 2 222
3 223
1 231
3 2 232
3 233
1 311
1 2 312
3 313
1 321
3 2 2 322
3 323
al 331
3 2 332
3 333
Effective, and not particularly difficult...but tedious. How could we have
answered without the table? Well, of course, it’s the rule of multiplication
again. There 3 possibilities for the first digit. For each of these, there are 3
possibilities for the second digit; and for each of these, 3 possibilities for
the third digit. 3 x 3 x 3 = 27.
Now, let’s ask a different problem: how many possible 3-digit numbers can
be made using the digits 1, 2, and 3, if every digit is used only once? Once
again, we can list them systematically—and it’s a lot easier this time. Once
you have chosen the first two digits, the third digit is forced. There are only
six possibilities.
First Digit Second Digit Third Digit Number
2 3 123
1
3 2 132
1 3 213
2
3 1 231
1 2 312
3
2 1 321
I really do believe it is important to show them these tables before doing
any calculations!!! There is no substitute for seeing everything laid out in
an organized manner to get a feeling for the space.
Once again, however, once they have seen the table, we can ask the
question: why 6? And once again, we can answer that question using the
rule of multiplication. There are three possible numbers that can go in the
first digit. Once you have chosen that digit, there are only two possible
numbers that can go in the second digit. And once you have chosen that,
there is only one number that can possibly go in the third. 3 x 2 x 1 = 6.
Exercise:
Problem:
Repeat the above problems, only with nine digits instead of three.
First, how many different nine-digit numbers can be made using the
digits 1-9? Second, how many different nine-digit numbers can be
made if you use the digits 1-9, but use each digit only once?
Obviously we don’t want to make these tables (even the second one is
prohibitive!) but with the rule of multiplication, and our calculators,
we can figure out how big the tables would be. Give them a couple of
minutes on this.
The firstis9 x 9x9x9x9x9x9x 9x Y. (Nine possibilities
for the first digit; for each of those, nine for the second; and so on.)
Even that is tedious to write. Let’s write it like this instead: 99. We can
punch it into the calculator just like that.
The secondis9 x 8x 7x6x5x4x 3x 2 x 1. (Nine possibilities
for the first digit; for each of those, only eight for the second, because
one of them is used up; and so on.) Is there any easy way to write that?
In fact, there is. It is called 9 factorial, and it is written 9! You may
want to show them how to get factorials on their calculators. On some
(such as the TI-83) the factorial option is actually listed under
probability, reflecting the fact that factorials are used so often in
probability problems, for this very reason.
Incidentally, 9° = 387,420,489 possibilities for the first scenario.
9! = 362,880 for the second: still a pretty big number, but only about
a thousandth as big as the first one. This should come as no surprise:
almost all of the nine-digit numbers use the same digit twice
somewhere or other!
Exercise:
Problem:
Question: How many different ways can five books be arranged on a
shelf? Give them a minute, and see if they can figure out that it is the
same problem we just did. 5 books can go in the first position; for each
of these, 4 in the second position; and so on. 5! = 120 possibilities.
Exercise:
Problem:
Question: How many three-digit numbers can be made using the digits
1-9?
If we are allowed to repeat digits, this is hopefully pretty easy by this
point: 9 x 9 x 9. Written more concisely, 9°.
But what if we’re not? Is there any way we can write 9 x 8 x 7 more
concisely? There is, and it’s a bit sneaky: it is 3. Explain why this
works. Point out that, while they may not particularly need it for
9 x 8 x 7, it’s really nice as a shortcut for 20 x 19 x 18...8.
If you have extra time, ask everyone in class to come up with two scenarios:
one of the “the same thing can be used twice” (exponential) variety, and one
of the “the same thing cannot be used twice” (factorial) variety.
Homework
“Homework: Permutations”
Probability -- Combinations
Once again, this one is lecture, without an in-class worksheet, walking
through a series of questions.
Suppose you have a Daisy, an Iris, a Lily, a Rose, and a Violet. You are
going to make a floral arrangement with three of them. How many possible
arrangements can you make?
Based on yesterday, it’s tempting to answer 5 x 4 x 3, or a But here’s
why this is different from yesterday’s problems: order doesn’t matter. “A
rose, a daisy, and an iris” is the same arrangement as “A daisy, an iris, and a
rose”: you don’t want to count it twice.
So, let’s start the way we did yesterday: list them all. Give the class a
minute to do this. Remind them, as always, that it’s best to be systematic to
make sure you list every possibility exactly once. Here’s my list.
First Second Third
Flower Flower Flower Arrangement
First Second Third
Flower Flower Flower Arrangement
Iris Lily Daisy, Iris, Lily
Rose Daisy, Iris, Rose
Daisy Violet Daisy, Iris, Violet
Lily
Rose Daisy, Lily, Rose
Violet Daisy, Lily, Violet
Rose Violet Daisy, Rose, Violet
First Second Third
Flower Flower Flower Arrangement
Iris
Lily
Rose Iris, Lily, Rose
Violet Iris, Lily, Violet
Rose Violet Iris, Rose, Violet
; : Lily, Rose,
Lily Rose Violet Violet
10 items in all. I generated the list by using the same kind of systematic
approach I used for the permutations, but always moving forward in the list:
so after “Lily” I’m allowed to list “Rose” and “Violet,” but not “Daisy” or
“Tris.”
This turns out to be such a common and important operation that it gets its
own name: “choose.” We would say “5 choose 3 is 10,” sometimes written
5
& = 10. It means, if you have five items, and want to choose 3 of them,
there are ten ways to do so.
Exercise:
Problem:
The four Beatles are John, Paul, Georg, and Ringo. Suppose you are
going to put photographs of three of them on your door. List all the
possible combinations. How many are there?
Give the class a minute to list—they should make a diagram like the one I
made above!—and count. When they are done, you can point out that the
answer (four combinations) is very obvious if you look at it backward: each
combination leaves exactly one Beatle out. There is a very important insight
here, which can also be applied to the flower example: each arrangement
listed leaves exactly two flowers out. So 5 choose 3 (“which flowers should
I include?”) is the same number as 5 choose 2 (“which flowers should I
leave out?”).
Now we’re going to try something with bigger numbers. A drama teacher
looks out at a class of 30 students, and wants to choose 2 of them to run a
scene. How many possible pairs of students are there? As before, when the
numbers get this large, we can’t reasonably list all the combinations: we
need an algebraic way of figuring out how many there are, without actually
counting them all.
To start off, let’s turn this combinations (“order doesn’t matter”) problem
into a permutations (“order does matter”) problem. The teacher wants to
choose one student to play the Child and one to play the Dog. In this
version of the problem, “John plays the Child and Susan plays the Dog” is
different from “Susan plays the Child and John plays the Dog,” and should
be counted separately. So it is a straightforward permutations problem.
There are 30 possible actors for the Child, and for each of those, 29 for the
Dog. 30 x 29 = 870 possible scenes.
Now let’s return to the original problem, how many pairs of students are
there? In this problem, “John-Susan” and “Susan-John” are the same pair.
The key insight here is that, when we ran the permutations problem, we
counted each pair twice. So the answer is 870/2 = 435 pairs.
Take this slow and easy. This is a very general approach to combinations
problem. First, you solve the (easier) permutations problem. Then you ask,
“How many times did I count every group?” and divide by that.
Ask the class to try this approach on the original (flower) problem. They
should answer three questions.
1. How many permutations are there? Remember that in this question,
“rose—daisy—iris” and “daisy—iris—rose” are two different
arrangements, as if each flower is being placed in a numbered slot.
2. Now, when we counted up the permutations, how many times did they
redundantly count each combination?
3. Divide the first answer by the second, and you have the total number
of combinations.
Lay out the process, then have them work the problem. Many of them will
get stuck on step (2), incorrectly thinking that we counted each permutation
three times. In fact, we counted each one six times:
daisy—iris—rose, daisy—rose—iris, iris—daisy—rose, iris—rose—daisy, rose—
daisy—iris, rose—iris—daisy
5!
97>
listed six times, so the total number of combinations is 10, as we counted
before.
The permutations are 5 x 4 x 3, or or 60. But each combination is
So...why six times? This is the last question, and it’s a hard one. When we
were choosing two items, we counted each combination twice (John—Susan,
Susan—John). When we were choosing three items, we counted each
combination six times. What’s the pattern? Give them a minute to think
about this. Then make sure they understand that it is, in fact...another
permutations problem! The list of six I gave above is simply the number of
ways you can arrange three items, or (3!). You would divide by (4!) for 4
items, and so on.
Homework
“Homework: Permutations and Combinations”
A sample test, and real test, and you’re done!
Live Music
Archive
Librivox
Free Audio
Metropolitan Museum
Cleveland
Museum of Art
Internet
Arcade
Console Living Room
Open Library
American
Libraries
TV News
Understanding
9/11